####################################################################
##SuckerBetsAnalysis.txt: Save this file as  SuckerBetsAnalysis.txt  #
## To use it, stay in the                                            #
##same directory, get into Maple (by typing: maple <Enter> )         #
##and then type:  read `SuckerBetsAnalysis.txt`<Enter>               #
##Then follow the instructions given there                           #
##                                                                   #
##Written by Doron Zeilberger, Rutgers University ,                  #
#DoronZeil at gmail dot com                                          #
######################################################################
 
#Created:  Sept.-Oct. 2017
 
with(combinat):

print(`Created:  Sept.-Oct.  2017`):
print(` This is SuckerBetsAnalysis.txt `):
print(`It is one of the Maple package that accompaniy the article `):
print(`A Treatise on Sucker's Bets`):
print(`by Shalosh B. Ekhad and Doron Zeilberger`):

print(`and also available from Zeilberger's website`):
print(``):
print(`Please report bugs to DoronZeil at gmail dot com `):
print(``):
 print(`The most current version of this  package and paper`):
 print(` are  available from`):
 print(`http://www.math.rutgers.edu/~zeilberg/  .`):

print(`---------------------------------------`):
 print(`For a list of the Supporting procedures type ezra1();, for help with`):
 print(`specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`---------------------------------------`):

print(`---------------------------------------`):
 print(`For a list of the Checking procedures type ezraC();, for help with`):
 print(`specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`---------------------------------------`):

print(`---------------------------------------`):
 print(`For a list of the Story procedures type ezraS();, for help with`):
 print(`specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`---------------------------------------`):

print(`---------------------------------------`):
 print(`For a list of the MAIN procedures type ezra();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`---------------------------------------`):

with(combinat):

ezraC:=proc()

if args=NULL then
 print(` The CHECKING procedures are:  CheckMomEZ, CheckMomEZd, CheckMomEZd1, CheckOpeD, CheckOpeE, CheckOpeO `):
 print(``):
else
ezra(args):
fi:

end:

ezraS:=proc()

if args=NULL then
 print(` The STORY procedures are:  MomForV, MomForV1, MomForVlatex, SBpaper, SBpaperLP  `):
 print(``):
else
ezra(args):
fi:

end:

ezra1:=proc()

if args=NULL then
 print(` The supporting procedures are:  BMdata, EZ, GP, GP1, , FMfor , GuessRat, Mijk, MijkS, OpeD, OpeD1,OpeE, OpeE1, OpeO, OpeO1, Pos, Sijk, SijkS,`):
 print(`TavlaLimSM,TavlaLimSMpc, TavlaMfor, TavlaMforPC `):
 print(` TavlaMom, TavlaMomc, TavlaMomcPC, TavlaMomPC, Trun `):
 print(``):

else
ezra(args):
fi:

end:

ezra:=proc()

if args=NULL then
 print(`The main procedures are:   BM, CMFdiag, GFq, LimSM, LimSMpc, Mfor, MomFor,  MomEZ, MomEZd, MomEZd1, ProbSucker, SBseq, VerifySBC  `):
 print(` `):


elif nops([args])=1 and op(1,[args])=BM then
print(`BM(a,b,c,t,R): the binomial moment generating function up to order R`):
print(`the coefficient of t[1]^i1*t[2]^i2*t[3]^i3 (for 0<=i1,i2,i3<=R) is`):
print(`the expectation of binomial(X1,i1)*binomial(X2,i2)*binomial(X3,i3).`):
print(`here X1 is the r.v. #of 12-#21, X2 is the r.v #23-32 , X3 is r.v. #31-#13`):
print(`Try: `):
print(`BM(3,3,3,t,2);`):

elif nops([args])=1 and op(1,[args])=BMdata then
print(`BMdata(i,j,k,R): collects data for the  binomial moments (i,j,k) for  GFq(n,n,n,q1,q2,q3) from n=1 to n=R. Try:`):
print(`BMdata(1,1,1,10);`):

elif nops([args])=1 and op(1,[args])=CheckMomEZ then
print(`CheckMomEZ(N): checks procedure MomEZ up to N (N<=16). Try:`):
print(`CheckMomEZ(10);`):

elif nops([args])=1 and op(1,[args])=CheckMomEZd then
print(`CheckMomEZd(N): checks procedure MomEZd up to N . Try:`):
print(`CheckMomEZd(20);`):

elif nops([args])=1 and op(1,[args])=CheckMomEZd1 then
print(`CheckMomEZd1(N): checks the explicit formula given by MomEZd1(n1) from n1=0 to n1=N. Try:`):
print(`CheckMomEZd1(20);`):



elif nops([args])=1 and op(1,[args])=CheckOpeE then
print(`CheckOpeE(): checks the validity of OpeE. Try:`):
print(`CheckOpeE();`):

elif nops([args])=1 and op(1,[args])=CheckOpeO then
print(`CheckOpeO(): checks the validity of OpeO. Try:`):
print(`CheckOpeO();`):

elif nops([args])=1 and op(1,[args])=CMFdiag then
print(`CMFdiag(a,b,c): the diagonal values of the tri-variate cumulative distribution function for the Sucker Bets statistics`):
print(`with three decks and deck-sizes a,b,c`):
print(` from [0,0,0], incremented by [1,1,1]. Try:`):
print(` CMFdiag(5,5,5); `):

elif nops([args])=1 and op(1,[args])=EZ then
print(`EZ(x,y,z,c):exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c):`):

elif nops([args])=1 and op(1,[args])=FMfor then
print(`FMfor(n,i,j,k): the formula for the (i,j,k) mixed factorial moment for our statistics. Try: `):
print(`FMfor(n,2,1,1);`):

elif nops([args])=1 and op(1,[args])=GFq then
print(`GFq(a,b,c,q): the Generating function according to q[1]^(#12-#21)*q[2]^(#23-#32)*q[2]^(#31-#13)`):
print(`of the set of all walks with unit steps from the origin to [a,b,c] in  the positive 3D rectangular lattice. Try:  `):
print(`GFq(3,3,3,q);`):

elif nops([args])=1 and op(1,[args])=GP then
print(`GP(L,n): inputs a list L of numbers and guesses a polynomial of degree <=nops(L)-3, or returns FAIL. Try:`):
print(`GP([seq(i^2,i=1..10)],n);`):

elif nops([args])=1 and op(1,[args])=GP1 then
print(`GP1(L,n,d): inputs a list L of numbers and guesses a polynomial of degree d. Try:`):
print(`GP1([seq(i^2,i=1..10)],n,2);`):

elif nops([args])=1 and op(1,[args])=GuessRat then
print(`GuessRat(gu,n): given a list gu `):
print(`guesses a rational function R(n) of degree <=d , such that R(gu[i][1])=gu[i][2] for all i`):
print(`For example, try: GuessRat([seq([i,i/(i+1)],i=1..20)],n);`):

elif nops([args])=1 and op(1,[args])=LimSM then
print(`LimSM(i,j,k): the limit, as n goes to infinite of the (i,j,k) scaled moment. Try: `):
print(`LimSM(2,2,2);`):

elif nops([args])=1 and op(1,[args])=LimSMpc then
print(`LimSMpc(i,j,k):Like  LimSM(i,j,k) (q.v.), but much faster, for max(i,j,k)<=5, since it uses pre-computed values. Try:`):
print(`LimSMpc(2,2,2);`):

elif nops([args])=1 and op(1,[args])=Mfor then
print(`Mfor(n,i,j,k): the formula for the (i,j,k) mixed  moment for our statistics. Try: `):
print(`Mfor(n,2,1,1);`):

elif nops([args])=1 and op(1,[args])=Mijk then
print(`Mijk(c,i,j,k): the (i,j,k) moment of exp(-x^2/2-y^2/-z^2/2 -(x*y+x*z+y*z)*c) normalized.`):
print(` Try: `):
print(`  Mijk(1/2,1,1,0) `):

elif nops([args])=1 and op(1,[args])=MijkS then
print(`MijkS(c,i,j,k): the (i,j,k) moment of exp(-x^2/2-y^2/-z^2/2 -(x*y+x*z+y*z)*c) normalized, for SYMBOLIC c, assumed to be between 0 and 1`):
print(` Try: `):
print(`  MijkS(c,2,1,1) `):


elif nops([args])=1 and op(1,[args])=MomEZ then
print(`MomEZ(b1,b2,b3,c): The (b1,b2,b3) scaled mixed moment of EZ(x,y,z,c) done via the recurrence obtained from the`):
print(`MultiAlmkvistZeilberger package of Apagodu and Zeilbegrger . Try:`):
print(`MomEZ(20,20,20,c);`):

elif nops([args])=1 and op(1,[args])=MomEZd then
print(`MomEZd(n1,c): The (2n1,2n1,2n1) scaled mixed moment of EZ(x,y,z,c) done via the diagonal recurrence obtained from the`):
print(`MultiAlmkvistZeilberger package of Apagodu and Zeilbegrger . Try:`):
print(`MomEZd(10,c);`):

elif nops([args])=1 and op(1,[args])=MomEZd1 then
print(`MomEZd1(n1) : the explicit formula (3*n1)!*(2*n1)!/8^n1/n1!^2 for the scaled mixed (2n1,2n1,2n1) Ekhad-Zeilbeger moment.`):
print(`Same as MomEZd(n1,1) but done via the explicit formula. Try:`):
print(`MomEZd1(10);`):

elif nops([args])=1 and op(1,[args])=MomFor then
print(`MomFor(N,k) inputs a pos. integer N and a variable name k, outputs a `):
print(`set of successful (a,b) and a table T, such that T[[a,b]] is`):
print(`an explicit formula, in k, for the (a,b,2*k) mixed moment for the trivariate continuous distribution lim(EZ(x,y,z,c),c->1) `);
print(` when a+b is even and for the (a,b,2*k-1) mixed moment if a+b is odd.`): 
print(`Try:`):
print(`MomFor(16,k);`):
elif nops([args])=1 and op(1,[args])=MomForV then
print(`MomForV(N,k): Verbose form of MomFor(N,k) (q.v). N is a pos. integer (up to 16, k is a variable name). Try:`):
print(`MomForV(15,k);`):

elif nops([args])=1 and op(1,[args])=MomForV1 then
print(`Like MomForV(N,k) but using lprint instead of print. Try:`):
print(`MomForV1(15,k);`):

elif nops([args])=1 and op(1,[args])=MomForVlatex then
print(`Like MomForV(N,k) but using latex instead of print. Try:`):
print(`MomForVlatex(15,k);`):

elif nops([args])=1 and op(1,[args])=OpeD then
print(`OpeD(n,N,c): The recurrence operator annihilating the scaled (2n,2n,2n) mixed moment  of the distribution,`):
print(`EZ(x,y,z,c)=exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c):`):
print(`where N is the shift operator in n.`):
print(` It was obtained via the Multivariable Almkvist-Zeilberger algorithm of Moa Apagodu and Doron Zeilberger. Try:`):
print(`OpeD(n,N,c):`):

elif nops([args])=1 and op(1,[args])=OpeD1 then
print(`opeD1(n,N,c): 1-N^(-2)OpeD(n,N,c). Try:`) :
print(`OpeD1(n,N,c);`):

elif nops([args])=1 and op(1,[args])=OpeE then
print(`opeE(a1,a2,a3,A3,c): The recurrence operator annihilating the scaled (a1,a2,2*a3) moment of the distribution`):
print(`EZ(x,y,z,c)=exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c).`):
print(` It was obtained via the Multivariable Almkvist-Zeilberger algorithm of Moa Apagodu and Doron Zeilberger. Try:`):
print(`OpeE(a1,a2,a3,A3,c);`):

elif nops([args])=1 and op(1,[args])=OpeE1 then
print(`opeE1(a1,a2,a3,A3,c): 1-A3^(-4)OpeE(a1,a2,a3,A3,c). Try:`) :
print(`OpeE1(a1,a2,a3,A3,c);`):

elif nops([args])=1 and op(1,[args])=OpeO then
print(`opeO(a1,a2,a3,A3,c): The recurrence operator annihilating the scaled (a1,a2,2*a3+1) moment of the distribution`):
print(`EZ(x,y,z,c)=exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c) .`):
print(` It was obtained via the Multivariable Almkvist-Zeilberger algorithm of Moa Apagodu and Doron Zeilberger. Try:`):
print(`OpeE(a1,a2,a3,A3,c);`):

elif nops([args])=1 and op(1,[args])=OpeO1 then
print(`opeO1(a1,a2,a3,A3,c): 1-A3^(-4)OpeO(a1,a2,a3,A3,c). Try:`):
print(`OpeO1(a1,a2,a3,A3,c);`):

elif nops([args])=1 and op(1,[args])=Pos then
print(`Pos(c): the integral of the distribution normalized exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c) over x,y,z>=0`):

elif nops([args])=1 and op(1,[args])=ProbSucker then
print(`ProbSucker(a,b,c): the probability that a random a-face die, b-faced die, and c-faced die, with all different labels on the`):
print(`a+b+c faces is  a sucker-bet's triple. Try:`):
print(`ProbSucker(3,3,3);`):

elif nops([args])=1 and op(1,[args])=SBseq then
print(`SBseq(N): the sequence of length N  whose n-th entry is the number of sucker bets with each of the decks with n cards`):
print(`or equivalently, the number of words w, in the alphabet {1,2,3} with n 1's n 2's and n 3's such that`):
print(`S12(w)>0, S23(w), and S31(w)>0. Try: `):
print(`SBseq(5);`):

elif nops([args])=1 and op(1,[args])=SBpaper then
print(`SBpaper(N): An article about the Sucker-Bets Statistics, giving all the mixed-moments up to the (N,N,N)`):
print(`and verifying that the limits of scaled limits are those of the corresponding one for EZ(x,y,z,c) as c goes to 1. Try:`):
print(` SBpaper(5);`):

elif nops([args])=1 and op(1,[args])=SBpaperLP then
print(`SBpaperLP(N): Like SBpaper(N) (q.v.) but using lprint instead of print. Try:`):
print(` SBpaperLP(5);`):

elif nops([args])=1 and op(1,[args])=Sijk then
print(`Sijk(c,i,j,k): the (i,j,k) standardized moment of exp(-x^2/2-y^2/-z^2/2 -(x*y+x*z+y*z)*c) normalized.`):
print(` Try: `):
print(`  Sijk(1/2,1,1,0) `):

elif nops([args])=1 and op(1,[args])=SijkS then
print(`SijkS(c,i,j,k): the (i,j,k) standardized moment of exp(-x^2/2-y^2/-z^2/2 -(x*y+x*z+y*z)*c) normalized. For symbolic c.`):
print(` Try: `):
print(`  SijkS(c,2,2,2) `):


elif nops([args])=1 and op(1,[args])=TavlaLimSM then
print(`TavlaLimSM(N): a list-of-lists-of-lists giving  LimSM`):
print(`for 0<=i,j,k<=N. Try:`):
print(`TavlaLimSM(2);`):

elif nops([args])=1 and op(1,[args])=TavlaLimSMpc then
print(`TavlaLimSMpc(N): a pre-computed version of TavlaLimSM(N), much faster for N<=5. Try:`):
print(`TavlaLimSMpc(2);`):


elif nops([args])=1 and op(1,[args])=TavlaMfor then
print(`TavlaMfor(N,n) a table of the mixed moments for the 3-Deck Sucker-Bets statistics with three decks and n cards in each`):
print(`(with all distinct 3*n cards). Try:`):
print(`TavlaMfor(2,n);`):

elif nops([args])=1 and op(1,[args])=TavlaMforPC then
print(`TavlaMfor(N,n) a table of the mixed moments for the 3-Deck Sucker-Bets statistics with three decks and n cards in each`):
print(`(with all distinct 3*n cards). Try:`):
print(`TavlaMforPC(2,n);`):

elif nops([args])=1 and op(1,[args])=TavlaMom then
print(`TavlaMom(N): a list-of-lists-of-lists giving SijkS(c,i,j,k) at c=1`):
print(`for 0<=i,j,k<=N. Try:`):
print(`TavlaMom(2);`):

elif nops([args])=1 and op(1,[args])=TavlaMomc then
print(`TavlaMomc(N,c): a list-of-lists-of-lists giving SijkS(c,i,j,k) `):
print(`for 0<=i,j,k<=N. Try:`):
print(`TavlaMomc(2,c);`):

elif nops([args])=1 and op(1,[args])=TavlaMomcPC then
print(`TavlaMomcPC(N,c): a pre-computed version of TavlaMomc(N,c) (q.v.) much faster for N<=16`):
print(`Try:`):
print(`TavlaMomcPC(16,c);`):

elif nops([args])=1 and op(1,[args])=TavlaMomPC then
print(`TavlaMomPC(N): pre-computed version of TavlaMom(N) with N<=16. Try:`):
print(`TavlaMomPC(16);`):

elif nops([args])=1 and op(1,[args])=Trun then
print(`Trun(F,var,R): given a function F in the list of variables var, finds its Taylor expansion to order R. Try: `):
print(` Trun(1/(1-x-y),[x,y],2); `):

elif nops([args])=1 and op(1,[args])=VerifySBC then
print(`VerifySBC(N): verifies (rigorously) that all the limits of the mixed moments, M(i,j,k)`):
print(` (for 0<=i,j,k<=N), of Sucker-Bets distribution with 3 decks and `):
print(` n cards in each deck (with all cards differently numbered) as n goes to infinity coincide with the corresponding `):
print(` mixed moments of the Ekhad-Zeilberger tri-variate distribution that is the limit as c->1 (from the left) of the  `):
print(` distribution whose joint p.df. is exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c): `):
print(` For N<=5 it is very fast, since it uses pre-computed values. For N>5 it is very slow. Try: `):
print(` VerifySBC(5); `):


else

print(`There is no ezra for`,args):
fi:
 
end:


#########start pre-computed
TavlaMomcPC:=proc(N,c) local gu,i1,j1,k1:
option remember:
gu:=
[[[1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0, 2027025], [0,
-c/(c+1), 0, -3*c/(c+1), 0, -15*c/(c+1), 0, -105*c/(c+1), 0, -945*c/(c+1), 0, -\
10395*c/(c+1), 0, -135135*c/(c+1), 0, -2027025*c/(c+1), 0], [1, 0, (3*c^2+2*c+1
)/(c+1)^2, 0, 3*(5*c^2+2*c+1)/(c+1)^2, 0, 15*(7*c^2+2*c+1)/(c+1)^2, 0, 105*(9*c
^2+2*c+1)/(c+1)^2, 0, 945*(11*c^2+2*c+1)/(c+1)^2, 0, 10395*(13*c^2+2*c+1)/(c+1)
^2, 0, 135135*(15*c^2+2*c+1)/(c+1)^2, 0, 2027025*(17*c^2+2*c+1)/(c+1)^2], [0, -\
3*c/(c+1), 0, -3*c*(5*c^2+6*c+3)/(c+1)^3, 0, -15*c*(7*c^2+6*c+3)/(c+1)^3, 0, -\
315*c*(3*c^2+2*c+1)/(c+1)^3, 0, -945*c*(11*c^2+6*c+3)/(c+1)^3, 0, -10395*c*(13*
c^2+6*c+3)/(c+1)^3, 0, -405405*c*(5*c^2+2*c+1)/(c+1)^3, 0, -2027025*c*(17*c^2+6
*c+3)/(c+1)^3, 0], [3, 0, 3*(5*c^2+2*c+1)/(c+1)^2, 0, 3*(35*c^4+60*c^3+42*c^2+
12*c+3)/(c+1)^4, 0, 45*(21*c^4+28*c^3+18*c^2+4*c+1)/(c+1)^4, 0, 315*(33*c^4+36*
c^3+22*c^2+4*c+1)/(c+1)^4, 0, 945*(143*c^4+132*c^3+78*c^2+12*c+3)/(c+1)^4, 0, 
31185*(65*c^4+52*c^3+30*c^2+4*c+1)/(c+1)^4, 0, 405405*(85*c^4+60*c^3+34*c^2+4*c
+1)/(c+1)^4, 0, 2027025*(323*c^4+204*c^3+114*c^2+12*c+3)/(c+1)^4], [0, -15*c/(c
+1), 0, -15*c*(7*c^2+6*c+3)/(c+1)^3, 0, -15*c*(63*c^4+140*c^3+130*c^2+60*c+15)/
(c+1)^5, 0, -315*c*(33*c^4+60*c^3+50*c^2+20*c+5)/(c+1)^5, 0, -945*c*(143*c^4+
220*c^3+170*c^2+60*c+15)/(c+1)^5, 0, -51975*c*(39*c^4+52*c^3+38*c^2+12*c+3)/(c+
1)^5, 0, -2027025*c*(17*c^4+20*c^3+14*c^2+4*c+1)/(c+1)^5, 0, -2027025*c*(323*c^
4+340*c^3+230*c^2+60*c+15)/(c+1)^5, 0], [15, 0, 15*(7*c^2+2*c+1)/(c+1)^2, 0, 45
*(21*c^4+28*c^3+18*c^2+4*c+1)/(c+1)^4, 0, 45*(231*c^6+630*c^5+735*c^4+460*c^3+
165*c^2+30*c+5)/(c+1)^6, 0, 315*(429*c^6+990*c^5+1035*c^4+580*c^3+195*c^2+30*c+
5)/(c+1)^6, 0, 14175*(143*c^6+286*c^5+275*c^4+140*c^3+45*c^2+6*c+1)/(c+1)^6, 0,
155925*(221*c^6+390*c^5+351*c^4+164*c^3+51*c^2+6*c+1)/(c+1)^6, 0, 2027025*(323*
c^6+510*c^5+435*c^4+188*c^3+57*c^2+6*c+1)/(c+1)^6, 0, 6081075*(2261*c^6+3230*c^
5+2635*c^4+1060*c^3+315*c^2+30*c+5)/(c+1)^6], [0, -105*c/(c+1), 0, -315*c*(3*c^
2+2*c+1)/(c+1)^3, 0, -315*c*(33*c^4+60*c^3+50*c^2+20*c+5)/(c+1)^5, 0, -315*c*(
429*c^6+1386*c^5+1953*c^4+1540*c^3+735*c^2+210*c+35)/(c+1)^7, 0, -2835*c*(715*c
^6+2002*c^5+2541*c^4+1820*c^3+805*c^2+210*c+35)/(c+1)^7, 0, -155925*c*(221*c^6+
546*c^5+637*c^4+420*c^3+175*c^2+42*c+7)/(c+1)^7, 0, -2027025*c*(323*c^6+714*c^5
+777*c^4+476*c^3+189*c^2+42*c+7)/(c+1)^7, 0, -42567525*c*(323*c^6+646*c^5+663*c
^4+380*c^3+145*c^2+30*c+5)/(c+1)^7, 0], [105, 0, 105*(9*c^2+2*c+1)/(c+1)^2, 0,
315*(33*c^4+36*c^3+22*c^2+4*c+1)/(c+1)^4, 0, 315*(429*c^6+990*c^5+1035*c^4+580*
c^3+195*c^2+30*c+5)/(c+1)^6, 0, 315*(6435*c^8+24024*c^7+39732*c^6+37800*c^5+
22610*c^4+8680*c^3+2100*c^2+280*c+35)/(c+1)^8, 0, 14175*(2431*c^8+8008*c^7+
12012*c^6+10472*c^5+5810*c^4+2072*c^3+476*c^2+56*c+7)/(c+1)^8, 0, 155925*(4199*
c^8+12376*c^7+17108*c^6+13832*c^5+7210*c^4+2408*c^3+532*c^2+56*c+7)/(c+1)^8, 0,
14189175*(969*c^8+2584*c^7+3332*c^6+2520*c^5+1246*c^4+392*c^3+84*c^2+8*c+1)/(c+
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945*c*(143*c^4+220*c^3+170*c^2+60*c+15)/(c+1)^5, 0, -2835*c*(715*c^6+2002*c^5+
2541*c^4+1820*c^3+805*c^2+210*c+35)/(c+1)^7, 0, -2835*c*(12155*c^8+51480*c^7+
97812*c^6+109032*c^5+78498*c^4+37800*c^3+12180*c^2+2520*c+315)/(c+1)^9, 0, -\
155925*c*(4199*c^8+15912*c^7+27612*c^6+28392*c^5+19026*c^4+8568*c^3+2604*c^2+
504*c+63)/(c+1)^9, 0, -6081075*c*(2261*c^8+7752*c^7+12444*c^6+11928*c^5+7518*c^
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c^6+31416*c^5+18774*c^4+7560*c^3+2100*c^2+360*c+45)/(c+1)^9, 0], [945, 0, 945*(
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8008*c^7+12012*c^6+10472*c^5+5810*c^4+2072*c^3+476*c^2+56*c+7)/(c+1)^8, 0, 
14175*(46189*c^10+218790*c^9+469755*c^8+600600*c^7+505890*c^6+293076*c^5+118230
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248625*c^8+294840*c^7+232050*c^6+126252*c^5+48090*c^4+12600*c^3+2205*c^2+210*c+
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22596*c^5+8190*c^4+2040*c^3+345*c^2+30*c+3)/(c+1)^10, 0, 212837625*(37145*c^10+
133722*c^9+229653*c^8+240312*c^7+169218*c^6+82908*c^5+28770*c^4+6840*c^3+1125*c
^2+90*c+9)/(c+1)^10], [0, -10395*c/(c+1), 0, -10395*c*(13*c^2+6*c+3)/(c+1)^3, 0
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27612*c^6+28392*c^5+19026*c^4+8568*c^3+2604*c^2+504*c+63)/(c+1)^9, 0, -155925*c
*(88179*c^10+461890*c^9+1106105*c^8+1595880*c^7+1540110*c^6+1043196*c^5+505890*
c^4+175560*c^3+42735*c^2+6930*c+693)/(c+1)^11, 0, -6081075*c*(52003*c^10+248710
*c^9+550715*c^8+740520*c^7+670230*c^6+427812*c^5+196350*c^4+64680*c^3+15015*c^2
+2310*c+231)/(c+1)^11, 0, -212837625*c*(37145*c^10+163438*c^9+337535*c^8+426360
*c^7+364650*c^6+220836*c^5+96558*c^4+30360*c^3+6765*c^2+990*c+99)/(c+1)^11, 0],
[10395, 0, 10395*(13*c^2+2*c+1)/(c+1)^2, 0, 31185*(65*c^4+52*c^3+30*c^2+4*c+1)/
(c+1)^4, 0, 155925*(221*c^6+390*c^5+351*c^4+164*c^3+51*c^2+6*c+1)/(c+1)^6, 0, 
155925*(4199*c^8+12376*c^7+17108*c^6+13832*c^5+7210*c^4+2408*c^3+532*c^2+56*c+7
)/(c+1)^8, 0, 467775*(29393*c^10+125970*c^9+248625*c^8+294840*c^7+232050*c^6+
126252*c^5+48090*c^4+12600*c^3+2205*c^2+210*c+21)/(c+1)^10, 0, 467775*(676039*c
^12+3879876*c^11+10253958*c^10+16482180*c^9+17934345*c^8+13909896*c^7+7882644*c
^6+3287592*c^5+1001385*c^4+217140*c^3+31878*c^2+2772*c+231)/(c+1)^12, 0, 
42567525*(185725*c^12+980628*c^11+2408934*c^10+3624060*c^9+3711015*c^8+2720520*
c^7+1462692*c^6+580536*c^5+168795*c^4+34980*c^3+4950*c^2+396*c+33)/(c+1)^12, 0,
638512875*(334305*c^12+1634380*c^11+3759074*c^10+5329500*c^9+5169615*c^8+
3603864*c^7+1848924*c^6+701976*c^5+195855*c^4+38940*c^3+5346*c^2+396*c+33)/(c+1
)^12], [0, -135135*c/(c+1), 0, -405405*c*(5*c^2+2*c+1)/(c+1)^3, 0, -2027025*c*(
17*c^4+20*c^3+14*c^2+4*c+1)/(c+1)^5, 0, -2027025*c*(323*c^6+714*c^5+777*c^4+476
*c^3+189*c^2+42*c+7)/(c+1)^7, 0, -6081075*c*(2261*c^8+7752*c^7+12444*c^6+11928*
c^5+7518*c^4+3192*c^3+924*c^2+168*c+21)/(c+1)^9, 0, -6081075*c*(52003*c^10+
248710*c^9+550715*c^8+740520*c^7+670230*c^6+427812*c^5+196350*c^4+64680*c^3+
15015*c^2+2310*c+231)/(c+1)^11, 0, -6081075*c*(1300075*c^12+8112468*c^11+
23455614*c^10+41570100*c^9+50358165*c^8+44015400*c^7+28549092*c^6+13909896*c^5+
5090085*c^4+1381380*c^3+270270*c^2+36036*c+3003)/(c+1)^13, 0, -638512875*c*(
334305*c^12+1931540*c^11+5215158*c^10+8683532*c^9+9930635*c^8+8226504*c^7+
5074212*c^6+2357784*c^5+824967*c^4+214500*c^3+40326*c^2+5148*c+429)/(c+1)^13, 0
], [135135, 0, 135135*(15*c^2+2*c+1)/(c+1)^2, 0, 405405*(85*c^4+60*c^3+34*c^2+4
*c+1)/(c+1)^4, 0, 2027025*(323*c^6+510*c^5+435*c^4+188*c^3+57*c^2+6*c+1)/(c+1)^
6, 0, 14189175*(969*c^8+2584*c^7+3332*c^6+2520*c^5+1246*c^4+392*c^3+84*c^2+8*c+
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22596*c^5+8190*c^4+2040*c^3+345*c^2+30*c+3)/(c+1)^10, 0, 42567525*(185725*c^12+
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14+33801950*c^13+106138123*c^12+205633428*c^11+274501227*c^10+266996730*c^9+
195089895*c^8+108763512*c^7+46483437*c^6+15153138*c^5+3708705*c^4+660660*c^3+
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179150335*c^12+327202876*c^11+413530117*c^10+382167786*c^9+266128863*c^8+
141772488*c^7+58032975*c^6+18156138*c^5+4273269*c^4+732732*c^3+87087*c^2+6006*c
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646*c^5+663*c^4+380*c^3+145*c^2+30*c+5)/(c+1)^7, 0, -42567525*c*(7429*c^8+23256
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212837625*c*(37145*c^10+163438*c^9+337535*c^8+426360*c^7+364650*c^6+220836*c^5+
96558*c^4+30360*c^3+6765*c^2+990*c+99)/(c+1)^11, 0, -638512875*c*(334305*c^12+
1931540*c^11+5215158*c^10+8683532*c^9+9930635*c^8+8226504*c^7+5074212*c^6+
2357784*c^5+824967*c^4+214500*c^3+40326*c^2+5148*c+429)/(c+1)^13, 0, -638512875
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781795014*c^9+635499865*c^8+399021480*c^7+195089895*c^6+74264190*c^5+21828807*c
^4+4864860*c^3+795795*c^2+90090*c+6435)/(c+1)^15, 0], [2027025, 0, 2027025*(17*
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6081075*(2261*c^6+3230*c^5+2635*c^4+1060*c^3+315*c^2+30*c+5)/(c+1)^6, 0, 
42567525*(7429*c^8+18088*c^7+21964*c^6+15640*c^5+7390*c^4+2200*c^3+460*c^2+40*c
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9694845*c^14+60843510*c^13+179150335*c^12+327202876*c^11+413530117*c^10+
382167786*c^9+266128863*c^8+141772488*c^7+58032975*c^6+18156138*c^5+4273269*c^4
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2326762800*c^15+8464602600*c^14+19199507600*c^13+30381192660*c^12+35555183664*c
^11+31827916120*c^10+22227313680*c^9+12237040530*c^8+5328145680*c^7+1828538712*
c^6+489368880*c^5+100120020*c^4+15135120*c^3+1595880*c^2+102960*c+6435)/(c+1)^
16]], [[0, -c/(c+1), 0, -3*c/(c+1), 0, -15*c/(c+1), 0, -105*c/(c+1), 0, -945*c/
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), 0, c*(c-1)/(c+1)^2, 0, 3*c*(3*c-1)/(c+1)^2, 0, 15*c*(5*c-1)/(c+1)^2, 0, 105*
c*(7*c-1)/(c+1)^2, 0, 945*c*(9*c-1)/(c+1)^2, 0, 10395*c*(11*c-1)/(c+1)^2, 0, 
135135*c*(13*c-1)/(c+1)^2, 0, 2027025*c*(15*c-1)/(c+1)^2], [0, c*(c-1)/(c+1)^2,
0, -3*c*(c^2+1)/(c+1)^3, 0, -15*c*(3*c^2+1)/(c+1)^3, 0, -105*c*(5*c^2+1)/(c+1)^
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c+1)^4, 0, 315*c*(15*c^3+5*c^2+5*c-1)/(c+1)^4, 0, 945*c*(77*c^3+21*c^2+21*c-3)/
(c+1)^4, 0, 31185*c*(39*c^3+9*c^2+9*c-1)/(c+1)^4, 0, 405405*c*(55*c^3+11*c^2+11
*c-1)/(c+1)^4, 0, 2027025*c*(221*c^3+39*c^2+39*c-3)/(c+1)^4], [0, 3*c*(3*c-1)/(
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)/(c+1)^5, 0, -10395*c*(91*c^4+56*c^3+42*c^2+3)/(c+1)^5, 0, -405405*c*(45*c^4+
24*c^3+18*c^2+1)/(c+1)^5, 0, -2027025*c*(187*c^4+88*c^3+66*c^2+3)/(c+1)^5, 0],
[-15*c/(c+1), 0, -15*c*(3*c^2+1)/(c+1)^3, 0, -15*c*(7*c^4+8*c^3+6*c^2+3)/(c+1)^
5, 0, 45*c*(21*c^5+35*c^4+30*c^3+10*c^2+5*c-5)/(c+1)^6, 0, 315*c*(99*c^5+135*c^
4+110*c^3+30*c^2+15*c-5)/(c+1)^6, 0, 4725*c*(143*c^5+165*c^4+130*c^3+30*c^2+15*
c-3)/(c+1)^6, 0, 155925*c*(91*c^5+91*c^4+70*c^3+14*c^2+7*c-1)/(c+1)^6, 0, 
2027025*c*(153*c^5+135*c^4+102*c^3+18*c^2+9*c-1)/(c+1)^6, 0, 2027025*c*(3553*c^
5+2805*c^4+2090*c^3+330*c^2+165*c-15)/(c+1)^6], [0, 15*c*(5*c-1)/(c+1)^2, 0, 45
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1)^6, 0, -315*c*(33*c^6+72*c^5+75*c^4+40*c^3+15*c^2+5)/(c+1)^7, 0, -2835*c*(143
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315*c*(429*c^7+1155*c^6+1449*c^5+1015*c^4+455*c^3+105*c^2+35*c-35)/(c+1)^8, 0,
14175*c*(429*c^7+1001*c^6+1155*c^5+735*c^4+315*c^3+63*c^2+21*c-7)/(c+1)^8, 0, 
155925*c*(1105*c^7+2275*c^6+2457*c^5+1435*c^4+595*c^3+105*c^2+35*c-7)/(c+1)^8,
0, 14189175*c*(323*c^7+595*c^6+609*c^5+329*c^4+133*c^3+21*c^2+7*c-1)/(c+1)^8, 0
, 42567525*c*(2907*c^7+4845*c^6+4743*c^5+2385*c^4+945*c^3+135*c^2+45*c-5)/(c+1)
^8], [0, 105*c*(7*c-1)/(c+1)^2, 0, 315*c*(15*c^3+5*c^2+5*c-1)/(c+1)^4, 0, 315*c
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1449*c^5+1015*c^4+455*c^3+105*c^2+35*c-35)/(c+1)^8, 0, -2835*c*(715*c^8+2288*c^
7+3388*c^6+2912*c^5+1610*c^4+560*c^3+140*c^2+35)/(c+1)^9, 0, -155925*c*(663*c^8
+1872*c^7+2548*c^6+2016*c^5+1050*c^4+336*c^3+84*c^2+7)/(c+1)^9, 0, -2027025*c*(
1615*c^8+4080*c^7+5180*c^6+3808*c^5+1890*c^4+560*c^3+140*c^2+7)/(c+1)^9, 0, -\
42567525*c*(2261*c^8+5168*c^7+6188*c^6+4256*c^5+2030*c^4+560*c^3+140*c^2+5)/(c+
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30*c^2+3)/(c+1)^5, 0, -2835*c*(143*c^6+264*c^5+255*c^4+120*c^3+45*c^2+5)/(c+1)^
7, 0, -2835*c*(715*c^8+2288*c^7+3388*c^6+2912*c^5+1610*c^4+560*c^3+140*c^2+35)/
(c+1)^9, 0, 14175*c*(2431*c^9+9009*c^8+15444*c^7+15708*c^6+10458*c^5+4662*c^4+
1428*c^3+252*c^2+63*c-63)/(c+1)^10, 0, 467775*c*(4199*c^9+13923*c^8+21996*c^7+
20748*c^6+12978*c^5+5418*c^4+1596*c^3+252*c^2+63*c-21)/(c+1)^10, 0, 42567525*c*
(1615*c^9+4845*c^8+7140*c^7+6300*c^6+3738*c^5+1470*c^4+420*c^3+60*c^2+15*c-3)/(
c+1)^10, 0, 42567525*c*(52003*c^9+142443*c^8+197676*c^7+164220*c^6+93114*c^5+
34650*c^4+9660*c^3+1260*c^2+315*c-45)/(c+1)^10], [0, 945*c*(9*c-1)/(c+1)^2, 0,
945*c*(77*c^3+21*c^2+21*c-3)/(c+1)^4, 0, 4725*c*(143*c^5+165*c^4+130*c^3+30*c^2
+15*c-3)/(c+1)^6, 0, 14175*c*(429*c^7+1001*c^6+1155*c^5+735*c^4+315*c^3+63*c^2+
21*c-7)/(c+1)^8, 0, 14175*c*(2431*c^9+9009*c^8+15444*c^7+15708*c^6+10458*c^5+
4662*c^4+1428*c^3+252*c^2+63*c-63)/(c+1)^10, 0, -155925*c*(4199*c^10+17680*c^9+
34515*c^8+40560*c^7+31710*c^6+17136*c^5+6510*c^4+1680*c^3+315*c^2+63)/(c+1)^11,
0, -6081075*c*(6783*c^10+25840*c^9+46665*c^8+51120*c^7+37590*c^6+19152*c^5+6930
*c^4+1680*c^3+315*c^2+21)/(c+1)^11, 0, -212837625*c*(7429*c^10+25840*c^9+43605*
c^8+44880*c^7+31290*c^6+15120*c^5+5250*c^4+1200*c^3+225*c^2+9)/(c+1)^11, 0], [-\
10395*c/(c+1), 0, -10395*c*(9*c^2+1)/(c+1)^3, 0, -10395*c*(91*c^4+56*c^3+42*c^2
+3)/(c+1)^5, 0, -155925*c*(65*c^6+104*c^5+95*c^4+40*c^3+15*c^2+1)/(c+1)^7, 0, -\
155925*c*(663*c^8+1872*c^7+2548*c^6+2016*c^5+1050*c^4+336*c^3+84*c^2+7)/(c+1)^9
, 0, -155925*c*(4199*c^10+17680*c^9+34515*c^8+40560*c^7+31710*c^6+17136*c^5+
6510*c^4+1680*c^3+315*c^2+63)/(c+1)^11, 0, 467775*c*(29393*c^11+138567*c^10+
303875*c^9+405405*c^8+364650*c^7+231462*c^6+105798*c^5+34650*c^4+8085*c^3+1155*
c^2+231*c-231)/(c+1)^12, 0, 42567525*c*(22287*c^11+95931*c^10+195415*c^9+244035
*c^8+206910*c^7+124278*c^6+54054*c^5+16830*c^4+3795*c^3+495*c^2+99*c-33)/(c+1)^
12, 0, 212837625*c*(185725*c^11+735471*c^10+1403435*c^9+1652145*c^8+1329570*c^7
+759990*c^6+316470*c^5+94050*c^4+20625*c^3+2475*c^2+495*c-99)/(c+1)^12], [0, 
10395*c*(11*c-1)/(c+1)^2, 0, 31185*c*(39*c^3+9*c^2+9*c-1)/(c+1)^4, 0, 155925*c*
(91*c^5+91*c^4+70*c^3+14*c^2+7*c-1)/(c+1)^6, 0, 155925*c*(1105*c^7+2275*c^6+
2457*c^5+1435*c^4+595*c^3+105*c^2+35*c-7)/(c+1)^8, 0, 467775*c*(4199*c^9+13923*
c^8+21996*c^7+20748*c^6+12978*c^5+5418*c^4+1596*c^3+252*c^2+63*c-21)/(c+1)^10,
0, 467775*c*(29393*c^11+138567*c^10+303875*c^9+405405*c^8+364650*c^7+231462*c^6
+105798*c^5+34650*c^4+8085*c^3+1155*c^2+231*c-231)/(c+1)^12, 0, -6081075*c*(
52003*c^12+271320*c^11+660858*c^10+987360*c^9+1005345*c^8+733392*c^7+392700*c^6
+155232*c^5+45045*c^4+9240*c^3+1386*c^2+231)/(c+1)^13, 0, -638512875*c*(37145*c
^12+178296*c^11+405042*c^10+568480*c^9+546975*c^8+378576*c^7+193116*c^6+72864*c
^5+20295*c^4+3960*c^3+594*c^2+33)/(c+1)^13, 0], [-135135*c/(c+1), 0, -135135*c*
(11*c^2+1)/(c+1)^3, 0, -405405*c*(45*c^4+24*c^3+18*c^2+1)/(c+1)^5, 0, -2027025*
c*(119*c^6+168*c^5+147*c^4+56*c^3+21*c^2+1)/(c+1)^7, 0, -2027025*c*(1615*c^8+
4080*c^7+5180*c^6+3808*c^5+1890*c^4+560*c^3+140*c^2+7)/(c+1)^9, 0, -6081075*c*(
6783*c^10+25840*c^9+46665*c^8+51120*c^7+37590*c^6+19152*c^5+6930*c^4+1680*c^3+
315*c^2+21)/(c+1)^11, 0, -6081075*c*(52003*c^12+271320*c^11+660858*c^10+987360*
c^9+1005345*c^8+733392*c^7+392700*c^6+155232*c^5+45045*c^4+9240*c^3+1386*c^2+
231)/(c+1)^13, 0, 42567525*c*(185725*c^13+1062347*c^12+2846922*c^11+4711278*c^
10+5360355*c^9+4420845*c^8+2716428*c^7+1257828*c^6+438867*c^5+113685*c^4+21450*
c^3+2574*c^2+429*c-429)/(c+1)^14, 0, 638512875*c*(1002915*c^13+5311735*c^12+
13327626*c^11+20785050*c^10+22401665*c^9+17568837*c^8+10301148*c^7+4562844*c^6+
1527669*c^5+379665*c^4+69498*c^3+7722*c^2+1287*c-429)/(c+1)^14], [0, 135135*c*(
13*c-1)/(c+1)^2, 0, 405405*c*(55*c^3+11*c^2+11*c-1)/(c+1)^4, 0, 2027025*c*(153*
c^5+135*c^4+102*c^3+18*c^2+9*c-1)/(c+1)^6, 0, 14189175*c*(323*c^7+595*c^6+609*c
^5+329*c^4+133*c^3+21*c^2+7*c-1)/(c+1)^8, 0, 42567525*c*(1615*c^9+4845*c^8+7140
*c^7+6300*c^6+3738*c^5+1470*c^4+420*c^3+60*c^2+15*c-3)/(c+1)^10, 0, 42567525*c*
(22287*c^11+95931*c^10+195415*c^9+244035*c^8+206910*c^7+124278*c^6+54054*c^5+
16830*c^4+3795*c^3+495*c^2+99*c-33)/(c+1)^12, 0, 42567525*c*(185725*c^13+
1062347*c^12+2846922*c^11+4711278*c^10+5360355*c^9+4420845*c^8+2716428*c^7+
1257828*c^6+438867*c^5+113685*c^4+21450*c^3+2574*c^2+429*c-429)/(c+1)^14, 0, -\
638512875*c*(334305*c^14+2080120*c^13+6084351*c^12+11051768*c^11+13902889*c^10+
12796784*c^9+8879871*c^8+4715568*c^7+1924923*c^6+600600*c^5+141141*c^4+24024*c^
3+3003*c^2+429)/(c+1)^15, 0], [-2027025*c/(c+1), 0, -2027025*c*(13*c^2+1)/(c+1)
^3, 0, -2027025*c*(187*c^4+88*c^3+66*c^2+3)/(c+1)^5, 0, -6081075*c*(969*c^6+
1224*c^5+1035*c^4+360*c^3+135*c^2+5)/(c+1)^7, 0, -42567525*c*(2261*c^8+5168*c^7
+6188*c^6+4256*c^5+2030*c^4+560*c^3+140*c^2+5)/(c+1)^9, 0, -212837625*c*(7429*c
^10+25840*c^9+43605*c^8+44880*c^7+31290*c^6+15120*c^5+5250*c^4+1200*c^3+225*c^2
+9)/(c+1)^11, 0, -638512875*c*(37145*c^12+178296*c^11+405042*c^10+568480*c^9+
546975*c^8+378576*c^7+193116*c^6+72864*c^5+20295*c^4+3960*c^3+594*c^2+33)/(c+1)
^13, 0, -638512875*c*(334305*c^14+2080120*c^13+6084351*c^12+11051768*c^11+
13902889*c^10+12796784*c^9+8879871*c^8+4715568*c^7+1924923*c^6+600600*c^5+
141141*c^4+24024*c^3+3003*c^2+429)/(c+1)^15, 0, 638512875*c*(9694845*c^15+
65189475*c^14+206711925*c^13+409003595*c^12+563904705*c^11+573251679*c^10+
443548105*c^9+265823415*c^8+124356375*c^7+45390345*c^6+12819807*c^5+2747745*c^4
+435435*c^3+45045*c^2+6435*c-6435)/(c+1)^16], [0, 2027025*c*(15*c-1)/(c+1)^2, 0
, 2027025*c*(221*c^3+39*c^2+39*c-3)/(c+1)^4, 0, 2027025*c*(3553*c^5+2805*c^4+
2090*c^3+330*c^2+165*c-15)/(c+1)^6, 0, 42567525*c*(2907*c^7+4845*c^6+4743*c^5+
2385*c^4+945*c^3+135*c^2+45*c-5)/(c+1)^8, 0, 42567525*c*(52003*c^9+142443*c^8+
197676*c^7+164220*c^6+93114*c^5+34650*c^4+9660*c^3+1260*c^2+315*c-45)/(c+1)^10,
0, 212837625*c*(185725*c^11+735471*c^10+1403435*c^9+1652145*c^8+1329570*c^7+
759990*c^6+316470*c^5+94050*c^4+20625*c^3+2475*c^2+495*c-99)/(c+1)^12, 0, 
638512875*c*(1002915*c^13+5311735*c^12+13327626*c^11+20785050*c^10+22401665*c^9
+17568837*c^8+10301148*c^7+4562844*c^6+1527669*c^5+379665*c^4+69498*c^3+7722*c^
2+1287*c-429)/(c+1)^14, 0, 638512875*c*(9694845*c^15+65189475*c^14+206711925*c^
13+409003595*c^12+563904705*c^11+573251679*c^10+443548105*c^9+265823415*c^8+
124356375*c^7+45390345*c^6+12819807*c^5+2747745*c^4+435435*c^3+45045*c^2+6435*c
-6435)/(c+1)^16, 0]], [[1, 0, (3*c^2+2*c+1)/(c+1)^2, 0, 3*(5*c^2+2*c+1)/(c+1)^2
, 0, 15*(7*c^2+2*c+1)/(c+1)^2, 0, 105*(9*c^2+2*c+1)/(c+1)^2, 0, 945*(11*c^2+2*c
+1)/(c+1)^2, 0, 10395*(13*c^2+2*c+1)/(c+1)^2, 0, 135135*(15*c^2+2*c+1)/(c+1)^2,
0, 2027025*(17*c^2+2*c+1)/(c+1)^2], [0, c*(c-1)/(c+1)^2, 0, -3*c*(c^2+1)/(c+1)^
3, 0, -15*c*(3*c^2+1)/(c+1)^3, 0, -105*c*(5*c^2+1)/(c+1)^3, 0, -945*c*(7*c^2+1)
/(c+1)^3, 0, -10395*c*(9*c^2+1)/(c+1)^3, 0, -135135*c*(11*c^2+1)/(c+1)^3, 0, -\
2027025*c*(13*c^2+1)/(c+1)^3, 0], [(3*c^2+2*c+1)/(c+1)^2, 0, -(c^3-9*c^2-3*c-1)
/(c+1)^3, 0, 3*(3*c^4+8*c^3+16*c^2+4*c+1)/(c+1)^4, 0, 15*(15*c^4+8*c^3+20*c^2+4
*c+1)/(c+1)^4, 0, 105*(35*c^4+8*c^3+24*c^2+4*c+1)/(c+1)^4, 0, 945*(63*c^4+8*c^3
+28*c^2+4*c+1)/(c+1)^4, 0, 10395*(99*c^4+8*c^3+32*c^2+4*c+1)/(c+1)^4, 0, 135135
*(143*c^4+8*c^3+36*c^2+4*c+1)/(c+1)^4, 0, 2027025*(195*c^4+8*c^3+40*c^2+4*c+1)/
(c+1)^4], [0, -3*c*(c^2+1)/(c+1)^3, 0, 3*c*(c^3-11*c^2-3*c-3)/(c+1)^4, 0, -15*c
*(3*c^4+14*c^3+22*c^2+6*c+3)/(c+1)^5, 0, -315*c*(5*c^4+6*c^3+10*c^2+2*c+1)/(c+1
)^5, 0, -945*c*(35*c^4+22*c^3+38*c^2+6*c+3)/(c+1)^5, 0, -10395*c*(63*c^4+26*c^3
+46*c^2+6*c+3)/(c+1)^5, 0, -405405*c*(33*c^4+10*c^3+18*c^2+2*c+1)/(c+1)^5, 0, -\
2027025*c*(143*c^4+34*c^3+62*c^2+6*c+3)/(c+1)^5, 0], [3*(5*c^2+2*c+1)/(c+1)^2,
0, 3*(3*c^4+8*c^3+16*c^2+4*c+1)/(c+1)^4, 0, -3*(5*c^5-71*c^4-78*c^3-78*c^2-15*c
-3)/(c+1)^5, 0, 45*(7*c^6+46*c^5+83*c^4+60*c^3+37*c^2+6*c+1)/(c+1)^6, 0, 315*(
45*c^6+86*c^5+135*c^4+68*c^3+43*c^2+6*c+1)/(c+1)^6, 0, 945*(385*c^6+426*c^5+617
*c^4+228*c^3+147*c^2+18*c+3)/(c+1)^6, 0, 31185*(273*c^6+214*c^5+295*c^4+84*c^3+
55*c^2+6*c+1)/(c+1)^6, 0, 405405*(495*c^6+302*c^5+403*c^4+92*c^3+61*c^2+6*c+1)/
(c+1)^6, 0, 2027025*(2431*c^6+1218*c^5+1589*c^4+300*c^3+201*c^2+18*c+3)/(c+1)^6
], [0, -15*c*(3*c^2+1)/(c+1)^3, 0, -15*c*(3*c^4+14*c^3+22*c^2+6*c+3)/(c+1)^5, 0
, 15*c*(7*c^5-123*c^4-210*c^3-190*c^2-45*c-15)/(c+1)^6, 0, -315*c*(9*c^6+76*c^5
+163*c^4+160*c^3+95*c^2+20*c+5)/(c+1)^7, 0, -945*c*(165*c^6+436*c^5+733*c^4+560
*c^3+335*c^2+60*c+15)/(c+1)^7, 0, -51975*c*(91*c^6+148*c^5+213*c^4+128*c^3+77*c
^2+12*c+3)/(c+1)^7, 0, -2027025*c*(63*c^6+76*c^5+99*c^4+48*c^3+29*c^2+4*c+1)/(c
+1)^7, 0, -2027025*c*(1683*c^6+1636*c^5+1993*c^4+800*c^3+485*c^2+60*c+15)/(c+1)
^7, 0], [15*(7*c^2+2*c+1)/(c+1)^2, 0, 15*(15*c^4+8*c^3+20*c^2+4*c+1)/(c+1)^4, 0
, 45*(7*c^6+46*c^5+83*c^4+60*c^3+37*c^2+6*c+1)/(c+1)^6, 0, -45*(21*c^7-441*c^6-\
1005*c^5-1075*c^4-565*c^3-255*c^2-35*c-5)/(c+1)^7, 0, 315*(99*c^8+1020*c^7+2574
*c^6+3120*c^5+2240*c^4+940*c^3+330*c^2+40*c+5)/(c+1)^8, 0, 14175*(143*c^8+484*c
^7+910*c^6+896*c^5+600*c^4+212*c^3+74*c^2+8*c+1)/(c+1)^8, 0, 155925*(455*c^8+
988*c^7+1526*c^6+1232*c^5+784*c^4+236*c^3+82*c^2+8*c+1)/(c+1)^8, 0, 2027025*(
1071*c^8+1780*c^7+2414*c^6+1632*c^5+1000*c^4+260*c^3+90*c^2+8*c+1)/(c+1)^8, 0,
6081075*(10659*c^8+14620*c^7+18126*c^6+10480*c^5+6240*c^4+1420*c^3+490*c^2+40*c
+5)/(c+1)^8], [0, -105*c*(5*c^2+1)/(c+1)^3, 0, -315*c*(5*c^4+6*c^3+10*c^2+2*c+1
)/(c+1)^5, 0, -315*c*(9*c^6+76*c^5+163*c^4+160*c^3+95*c^2+20*c+5)/(c+1)^7, 0, 
315*c*(33*c^7-807*c^6-2289*c^5-2933*c^4-2065*c^3-945*c^2-175*c-35)/(c+1)^8, 0,
-2835*c*(143*c^8+1738*c^7+5076*c^6+7294*c^5+6286*c^4+3430*c^3+1260*c^2+210*c+35
)/(c+1)^9, 0, -155925*c*(195*c^8+806*c^7+1712*c^6+2030*c^5+1582*c^4+770*c^3+280
*c^2+42*c+7)/(c+1)^9, 0, -2027025*c*(595*c^8+1626*c^7+2780*c^6+2758*c^5+1974*c^
4+854*c^3+308*c^2+42*c+7)/(c+1)^9, 0, -42567525*c*(969*c^8+2074*c^7+3064*c^6+
2602*c^5+1738*c^4+670*c^3+240*c^2+30*c+5)/(c+1)^9, 0], [105*(9*c^2+2*c+1)/(c+1)
^2, 0, 105*(35*c^4+8*c^3+24*c^2+4*c+1)/(c+1)^4, 0, 315*(45*c^6+86*c^5+135*c^4+
68*c^3+43*c^2+6*c+1)/(c+1)^6, 0, 315*(99*c^8+1020*c^7+2574*c^6+3120*c^5+2240*c^
4+940*c^3+330*c^2+40*c+5)/(c+1)^8, 0, -315*(429*c^9-11979*c^8-40572*c^7-61292*c
^6-53130*c^5-29610*c^4-10220*c^3-2940*c^2-315*c-35)/(c+1)^9, 0, 14175*(429*c^10
+6006*c^9+19987*c^8+33208*c^7+33586*c^6+22260*c^5+10038*c^4+2968*c^3+721*c^2+70
*c+7)/(c+1)^10, 0, 155925*(3315*c^10+16198*c^9+38779*c^8+53480*c^7+48342*c^6+
29092*c^5+12446*c^4+3304*c^3+791*c^2+70*c+7)/(c+1)^10, 0, 14189175*(1615*c^10+
5338*c^9+10197*c^8+11880*c^7+9702*c^6+5292*c^5+2170*c^4+520*c^3+123*c^2+10*c+1)
/(c+1)^10, 0, 42567525*(20349*c^10+53618*c^9+87873*c^8+88728*c^7+66474*c^6+
32940*c^5+13050*c^4+2840*c^3+665*c^2+50*c+5)/(c+1)^10], [0, -945*c*(7*c^2+1)/(c
+1)^3, 0, -945*c*(35*c^4+22*c^3+38*c^2+6*c+3)/(c+1)^5, 0, -945*c*(165*c^6+436*c
^5+733*c^4+560*c^3+335*c^2+60*c+15)/(c+1)^7, 0, -2835*c*(143*c^8+1738*c^7+5076*
c^6+7294*c^5+6286*c^4+3430*c^3+1260*c^2+210*c+35)/(c+1)^9, 0, 2835*c*(715*c^9-\
22451*c^8-88308*c^7-154428*c^6-158550*c^5-106218*c^4-47460*c^3-14700*c^2-2205*c
-315)/(c+1)^10, 0, -155925*c*(663*c^10+10504*c^9+39299*c^8+74160*c^7+86286*c^6+
67200*c^5+36414*c^4+13776*c^3+3675*c^2+504*c+63)/(c+1)^11, 0, -6081075*c*(1615*
c^10+9112*c^9+24425*c^8+38352*c^7+39702*c^6+28224*c^5+14322*c^4+5040*c^3+1323*c
^2+168*c+21)/(c+1)^11, 0, -42567525*c*(11305*c^10+43928*c^9+93721*c^8+124944*c^
7+115986*c^6+75264*c^5+35994*c^4+11760*c^3+3045*c^2+360*c+45)/(c+1)^11, 0], [
945*(11*c^2+2*c+1)/(c+1)^2, 0, 945*(63*c^4+8*c^3+28*c^2+4*c+1)/(c+1)^4, 0, 945*
(385*c^6+426*c^5+617*c^4+228*c^3+147*c^2+18*c+3)/(c+1)^6, 0, 14175*(143*c^8+484
*c^7+910*c^6+896*c^5+600*c^4+212*c^3+74*c^2+8*c+1)/(c+1)^8, 0, 14175*(429*c^10+
6006*c^9+19987*c^8+33208*c^7+33586*c^6+22260*c^5+10038*c^4+2968*c^3+721*c^2+70*
c+7)/(c+1)^10, 0, -14175*(2431*c^11-84799*c^10-379665*c^9-756195*c^8-897330*c^7
-705726*c^6-382746*c^5-145950*c^4-37485*c^3-7875*c^2-693*c-63)/(c+1)^11, 0, 
467775*(4199*c^12+74256*c^11+308828*c^10+652860*c^9+859935*c^8+769776*c^7+
488544*c^6+223272*c^5+73605*c^4+16800*c^3+3108*c^2+252*c+21)/(c+1)^12, 0, 
42567525*(4845*c^12+31008*c^11+92344*c^10+162660*c^9+190305*c^8+155760*c^7+
92232*c^6+39576*c^5+12495*c^4+2640*c^3+480*c^2+36*c+3)/(c+1)^12, 0, 212837625*(
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144648*c^5+44025*c^4+8640*c^3+1548*c^2+108*c+9)/(c+1)^12], [0, -10395*c*(9*c^2+
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1809665*c^8-2438370*c^7-2206314*c^6-1405866*c^5-644490*c^4-211365*c^3-49665*c^2
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c^2+2310*c+231)/(c+1)^13, 0, -212837625*c*(22287*c^12+159562*c^11+523906*c^10+
1023638*c^9+1337765*c^8+1238820*c^7+840972*c^6+423852*c^5+159753*c^4+43890*c^3+
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0, 42567525*(22287*c^14+476102*c^13+2381479*c^12+6123876*c^11+9967419*c^10+
11259138*c^9+9260163*c^8+5686296*c^7+2635149*c^6+920106*c^5+239349*c^4+44484*c^
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10197*c^8+11880*c^7+9702*c^6+5292*c^5+2170*c^4+520*c^3+123*c^2+10*c+1)/(c+1)^10
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22287*c^14+476102*c^13+2381479*c^12+6123876*c^11+9967419*c^10+11259138*c^9+
9260163*c^8+5686296*c^7+2635149*c^6+920106*c^5+239349*c^4+44484*c^3+6633*c^2+
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179855767*c^12-330044715*c^11-417714297*c^10-386026641*c^9-268634223*c^8-\
142958673*c^7-58453395*c^6-18261243*c^5-4297293*c^4-729729*c^3-99099*c^2-6435*c
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445597880*c^13+864918418*c^12+1181592412*c^11+1195892698*c^10+923551200*c^9+
553238400*c^8+258901500*c^7+94588494*c^6+26738712*c^5+5747742*c^4+900900*c^3+
112398*c^2+6864*c+429)/(c+1)^16], [0, -2027025*c*(13*c^2+1)/(c+1)^3, 0, -\
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3064*c^6+2602*c^5+1738*c^4+670*c^3+240*c^2+30*c+5)/(c+1)^9, 0, -42567525*c*(
11305*c^10+43928*c^9+93721*c^8+124944*c^7+115986*c^6+75264*c^5+35994*c^4+11760*
c^3+3045*c^2+360*c+45)/(c+1)^11, 0, -212837625*c*(22287*c^12+159562*c^11+523906
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251336085*c^6-91858767*c^5-25972947*c^4-5570565*c^3-885885*c^2-83655*c-6435)/(c
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490*c^2+40*c+5)/(c+1)^8, 0, 42567525*(20349*c^10+53618*c^9+87873*c^8+88728*c^7+
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445597880*c^13+864918418*c^12+1181592412*c^11+1195892698*c^10+923551200*c^9+
553238400*c^8+258901500*c^7+94588494*c^6+26738712*c^5+5747742*c^4+900900*c^3+
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4176583800*c^15-14575995160*c^14-31535749700*c^13-47592322596*c^12-53189560424*
c^11-45548880520*c^10-30484950210*c^9-16112695170*c^8-6746450568*c^7-2229979752
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0, 155925*c*(273*c^7+651*c^6+1029*c^5+623*c^4+379*c^3+41*c^2+15*c-3)/(c+1)^8, 0
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0, 6081075*c*(7293*c^7+9251*c^6+13453*c^5+5995*c^4+3695*c^3+385*c^2+135*c-15)/(
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0, 945*c*(3*c^7-65*c^6-107*c^5-115*c^4-35*c^3-55*c^2-5*c-5)/(c+1)^8, 0, -2835*c
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0, -945*c*(33*c^8-828*c^7-1834*c^6-2184*c^5-1260*c^4-700*c^3+210*c^2+35)/(c+1)^
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42*c^2+21*c-21)/(c+1)^10, 0, 467775*c*(455*c^9+2847*c^8+6390*c^7+7434*c^6+5572*
c^5+2520*c^4+938*c^3+70*c^2+21*c-7)/(c+1)^10, 0, 42567525*c*(255*c^9+961*c^8+
1870*c^7+1918*c^6+1372*c^5+560*c^4+210*c^3+18*c^2+5*c-1)/(c+1)^10, 0, 42567525*
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1260*c^4-700*c^3+210*c^2+35)/(c+1)^9, 0, 2835*c*(143*c^9-4081*c^8-11316*c^7-\
15540*c^6-11886*c^5-6510*c^4-1540*c^3-1540*c^2-105*c-105)/(c+1)^10, 0, -155925*
c*(195*c^10+4394*c^9+13753*c^8+21240*c^7+19950*c^6+12348*c^5+5250*c^4+1400*c^3+
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4845*c^10+21726*c^9+47451*c^8+57672*c^7+47250*c^6+24948*c^5+10150*c^4+2120*c^3+
625*c^2+30*c+15)/(c+1)^11, 0], [-945*c*(11*c^2+6*c+3)/(c+1)^3, 0, -945*c*(35*c^
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1105*c^11+28119*c^10+100347*c^9+177189*c^8+192618*c^7+140742*c^6+71862*c^5+
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11-722436*c^10-1475760*c^9-1834305*c^8-1549944*c^7-924000*c^6-402864*c^5-121275
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18443997*c^7-7216209*c^6-2153151*c^5-495495*c^4-69069*c^3-33033*c^2-1287*c-1287
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354534180*c^8-159116100*c^7-55777722*c^6-15207192*c^5-3093090*c^4-540540*c^3+
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)/(c+1)^5, 0, -2027025*c*(323*c^4+340*c^3+230*c^2+60*c+15)/(c+1)^5, 0], [-15*c/
(c+1), 0, -15*c*(3*c^2+1)/(c+1)^3, 0, -15*c*(7*c^4+8*c^3+6*c^2+3)/(c+1)^5, 0, 
45*c*(21*c^5+35*c^4+30*c^3+10*c^2+5*c-5)/(c+1)^6, 0, 315*c*(99*c^5+135*c^4+110*
c^3+30*c^2+15*c-5)/(c+1)^6, 0, 4725*c*(143*c^5+165*c^4+130*c^3+30*c^2+15*c-3)/(
c+1)^6, 0, 155925*c*(91*c^5+91*c^4+70*c^3+14*c^2+7*c-1)/(c+1)^6, 0, 2027025*c*(
153*c^5+135*c^4+102*c^3+18*c^2+9*c-1)/(c+1)^6, 0, 2027025*c*(3553*c^5+2805*c^4+
2090*c^3+330*c^2+165*c-15)/(c+1)^6], [0, -15*c*(3*c^2+1)/(c+1)^3, 0, -15*c*(3*c
^4+14*c^3+22*c^2+6*c+3)/(c+1)^5, 0, 15*c*(7*c^5-123*c^4-210*c^3-190*c^2-45*c-15
)/(c+1)^6, 0, -315*c*(9*c^6+76*c^5+163*c^4+160*c^3+95*c^2+20*c+5)/(c+1)^7, 0, -\
945*c*(165*c^6+436*c^5+733*c^4+560*c^3+335*c^2+60*c+15)/(c+1)^7, 0, -51975*c*(
91*c^6+148*c^5+213*c^4+128*c^3+77*c^2+12*c+3)/(c+1)^7, 0, -2027025*c*(63*c^6+76
*c^5+99*c^4+48*c^3+29*c^2+4*c+1)/(c+1)^7, 0, -2027025*c*(1683*c^6+1636*c^5+1993
*c^4+800*c^3+485*c^2+60*c+15)/(c+1)^7, 0], [-15*c*(7*c^2+6*c+3)/(c+1)^3, 0, -15
*c*(3*c^4+14*c^3+22*c^2+6*c+3)/(c+1)^5, 0, 45*c*(c^5-15*c^4-6*c^3-22*c^2-3*c-3)
/(c+1)^6, 0, -45*c*(7*c^6-128*c^5-135*c^4-160*c^3+65*c^2+15)/(c+1)^7, 0, 315*c*
(45*c^7+619*c^6+1157*c^5+1035*c^4+575*c^3+25*c^2+15*c-15)/(c+1)^8, 0, 14175*c*(
77*c^7+321*c^6+543*c^5+395*c^4+235*c^3+23*c^2+9*c-3)/(c+1)^8, 0, 155925*c*(273*
c^7+651*c^6+1029*c^5+623*c^4+379*c^3+41*c^2+15*c-3)/(c+1)^8, 0, 2027025*c*(693*
c^7+1149*c^6+1731*c^5+891*c^4+547*c^3+59*c^2+21*c-3)/(c+1)^8, 0, 6081075*c*(
7293*c^7+9251*c^6+13453*c^5+5995*c^4+3695*c^3+385*c^2+135*c-15)/(c+1)^8], [0, -\
15*c*(7*c^4+8*c^3+6*c^2+3)/(c+1)^5, 0, 45*c*(c^5-15*c^4-6*c^3-22*c^2-3*c-3)/(c+
1)^6, 0, -45*c*(3*c^6-34*c^5+273*c^4+180*c^3+205*c^2+30*c+15)/(c+1)^7, 0, 315*c
*(5*c^7-113*c^6-447*c^5-869*c^4-505*c^3-315*c^2-45*c-15)/(c+1)^8, 0, -2835*c*(
35*c^8+660*c^7+1552*c^6+2292*c^5+2118*c^4+1020*c^3+440*c^2+60*c+15)/(c+1)^9, 0,
-155925*c*(63*c^8+364*c^7+688*c^6+724*c^5+614*c^4+244*c^3+104*c^2+12*c+3)/(c+1)
^9, 0, -2027025*c*(231*c^8+756*c^7+1296*c^6+1060*c^5+846*c^4+284*c^3+120*c^2+12
*c+3)/(c+1)^9, 0, -6081075*c*(3003*c^8+6732*c^7+10928*c^6+7332*c^5+5598*c^4+
1620*c^3+680*c^2+60*c+15)/(c+1)^9, 0], [-15*c*(63*c^4+140*c^3+130*c^2+60*c+15)/
(c+1)^5, 0, 15*c*(7*c^5-123*c^4-210*c^3-190*c^2-45*c-15)/(c+1)^6, 0, -45*c*(3*c
^6-34*c^5+273*c^4+180*c^3+205*c^2+30*c+15)/(c+1)^7, 0, 225*c*(c-1)*(3*c^6-34*c^
5+273*c^4+180*c^3+205*c^2+30*c+15)/(c+1)^8, 0, -1575*c*(7*c^8-192*c^7-1164*c^6-\
1952*c^5-942*c^4-480*c^3+100*c^2+15)/(c+1)^9, 0, 14175*c*(63*c^9+1505*c^8+4460*
c^7+7700*c^6+7130*c^5+3462*c^4+1260*c^3+20*c^2+15*c-15)/(c+1)^10, 0, 155925*c*(
693*c^9+5145*c^8+11340*c^7+14940*c^6+13270*c^5+5902*c^4+2300*c^3+140*c^2+45*c-\
15)/(c+1)^10, 0, 2027025*c*(3003*c^9+12705*c^8+24300*c^7+25540*c^6+21730*c^5+
8742*c^4+3500*c^3+260*c^2+75*c-15)/(c+1)^10, 0, 30405375*c*(9009*c^9+26169*c^8+
46156*c^7+40204*c^6+32926*c^5+11982*c^4+4860*c^3+380*c^2+105*c-15)/(c+1)^10], [
0, 45*c*(21*c^5+35*c^4+30*c^3+10*c^2+5*c-5)/(c+1)^6, 0, -45*c*(7*c^6-128*c^5-\
135*c^4-160*c^3+65*c^2+15)/(c+1)^7, 0, 225*c*(c-1)*(3*c^6-34*c^5+273*c^4+180*c^
3+205*c^2+30*c+15)/(c+1)^8, 0, -4725*c*(c^8-14*c^7+140*c^6+118*c^5+298*c^4+110*
c^3+100*c^2+10*c+5)/(c+1)^9, 0, 14175*c*(7*c^9-227*c^8-1824*c^7-3504*c^6-3234*c
^5-2598*c^4-920*c^3-440*c^2-45*c-15)/(c+1)^10, 0, -155925*c*(63*c^10+1820*c^9+
6595*c^8+12960*c^7+14190*c^6+10152*c^5+5550*c^4+1760*c^3+595*c^2+60*c+15)/(c+1)
^11, 0, -6081075*c*(231*c^10+2100*c^9+5425*c^8+8400*c^7+8350*c^6+5144*c^5+2650*
c^4+720*c^3+235*c^2+20*c+5)/(c+1)^11, 0, -30405375*c*(3003*c^10+15708*c^9+34035
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, 0], [-315*c*(33*c^4+60*c^3+50*c^2+20*c+5)/(c+1)^5, 0, -315*c*(9*c^6+76*c^5+
163*c^4+160*c^3+95*c^2+20*c+5)/(c+1)^7, 0, 315*c*(5*c^7-113*c^6-447*c^5-869*c^4
-505*c^3-315*c^2-45*c-15)/(c+1)^8, 0, -4725*c*(c^8-14*c^7+140*c^6+118*c^5+298*c
^4+110*c^3+100*c^2+10*c+5)/(c+1)^9, 0, 4725*c*(9*c^9-143*c^8+1696*c^7+2632*c^6+
3682*c^5+154*c^4+280*c^3-560*c^2-35*c-35)/(c+1)^10, 0, -14175*c*(77*c^10-2888*c
^9-28755*c^8-64920*c^7-75390*c^6-56952*c^5-19222*c^4-6440*c^3+945*c^2+105)/(c+1
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54838*c^5+18634*c^4+5005*c^3+35*c^2+35*c-35)/(c+1)^12, 0, 42567525*c*(495*c^11+
5331*c^10+16003*c^9+28335*c^8+32030*c^7+24150*c^6+13454*c^5+4342*c^4+1235*c^3+
55*c^2+15*c-5)/(c+1)^12, 0, 212837625*c*(7293*c^11+45639*c^10+112335*c^9+165789
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12], [0, 315*c*(99*c^5+135*c^4+110*c^3+30*c^2+15*c-5)/(c+1)^6, 0, 315*c*(45*c^7
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192*c^7-1164*c^6-1952*c^5-942*c^4-480*c^3+100*c^2+15)/(c+1)^9, 0, 4725*c*(9*c^9
-143*c^8+1696*c^7+2632*c^6+3682*c^5+154*c^4+280*c^3-560*c^2-35*c-35)/(c+1)^10,
0, -14175*c*(33*c^10-590*c^9+8117*c^8+17784*c^7+24402*c^6+14028*c^5+13986*c^4+
3640*c^3+2765*c^2+210*c+105)/(c+1)^11, 0, 155925*c*(91*c^11-3879*c^10-45975*c^9
-120925*c^8-166050*c^7-143766*c^6-79646*c^5-39690*c^4-10185*c^3-3955*c^2-315*c-\
105)/(c+1)^12, 0, -6081075*c*(315*c^12+12236*c^11+61238*c^10+152540*c^9+223165*
c^8+217240*c^7+145684*c^6+69272*c^5+26565*c^4+6300*c^3+1750*c^2+140*c+35)/(c+1)
^13, 0, -212837625*c*(1683*c^12+20964*c^11+72234*c^10+144108*c^9+184425*c^8+
161832*c^7+102396*c^6+44440*c^5+16213*c^4+3380*c^3+890*c^2+60*c+15)/(c+1)^13, 0
], [-945*c*(143*c^4+220*c^3+170*c^2+60*c+15)/(c+1)^5, 0, -945*c*(165*c^6+436*c^
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-1824*c^7-3504*c^6-3234*c^5-2598*c^4-920*c^3-440*c^2-45*c-15)/(c+1)^10, 0, -\
14175*c*(33*c^10-590*c^9+8117*c^8+17784*c^7+24402*c^6+14028*c^5+13986*c^4+3640*
c^3+2765*c^2+210*c+105)/(c+1)^11, 0, 42525*c*(143*c^11-2849*c^10+44633*c^9+
125145*c^8+188310*c^7+152502*c^6+102354*c^5+10962*c^4+5355*c^3-6405*c^2-315*c-\
315)/(c+1)^12, 0, -467775*c*(455*c^12-21736*c^11-298494*c^10-901728*c^9-1426515
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^13, 0, 42567525*c*(765*c^13+33515*c^12+191386*c^11+531006*c^10+875619*c^9+
965805*c^8+747180*c^7+419844*c^6+173139*c^5+45621*c^4+9690*c^3+30*c^2+45*c-45)/
(c+1)^14, 0, 638512875*c*(10659*c^13+150807*c^12+589114*c^11+1310890*c^10+
1881793*c^9+1876677*c^8+1353852*c^7+709116*c^6+285237*c^5+72561*c^4+16170*c^3+
570*c^2+135*c-45)/(c+1)^14], [0, 4725*c*(143*c^5+165*c^4+130*c^3+30*c^2+15*c-3)
/(c+1)^6, 0, 14175*c*(77*c^7+321*c^6+543*c^5+395*c^4+235*c^3+23*c^2+9*c-3)/(c+1
)^8, 0, 14175*c*(63*c^9+1505*c^8+4460*c^7+7700*c^6+7130*c^5+3462*c^4+1260*c^3+
20*c^2+15*c-15)/(c+1)^10, 0, -14175*c*(77*c^10-2888*c^9-28755*c^8-64920*c^7-\
75390*c^6-56952*c^5-19222*c^4-6440*c^3+945*c^2+105)/(c+1)^11, 0, 42525*c*(143*c
^11-2849*c^10+44633*c^9+125145*c^8+188310*c^7+152502*c^6+102354*c^5+10962*c^4+
5355*c^3-6405*c^2-315*c-315)/(c+1)^12, 0, -2338875*c*(39*c^12-858*c^11+15082*c^
10+51334*c^9+86865*c^8+85836*c^7+61068*c^6+25788*c^5+15561*c^4+3150*c^3+2058*c^
2+126*c+63)/(c+1)^13, 0, 30405375*c*(119*c^13-6299*c^12-98286*c^11-336354*c^10-\
603051*c^9-686889*c^8-538836*c^7-305004*c^6-122031*c^5-43869*c^4-8862*c^3-2898*
c^2-189*c-63)/(c+1)^14, 0, -638512875*c*(969*c^14+47260*c^13+303495*c^12+931040
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60669*c^4+11520*c^3+2715*c^2+180*c+45)/(c+1)^15, 0], [-51975*c*(39*c^4+52*c^3+
38*c^2+12*c+3)/(c+1)^5, 0, -51975*c*(91*c^6+148*c^5+213*c^4+128*c^3+77*c^2+12*c
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3879*c^10-45975*c^9-120925*c^8-166050*c^7-143766*c^6-79646*c^5-39690*c^4-10185*
c^3-3955*c^2-315*c-105)/(c+1)^12, 0, -2338875*c*(39*c^12-858*c^11+15082*c^10+
51334*c^9+86865*c^8+85836*c^7+61068*c^6+25788*c^5+15561*c^4+3150*c^3+2058*c^2+
126*c+63)/(c+1)^13, 0, 2338875*c*(663*c^13-15977*c^12+311402*c^11+1244298*c^10+
2370621*c^9+2725437*c^8+2184732*c^7+1198428*c^6+537537*c^5+64449*c^4+19866*c^3-\
17094*c^2-693*c-693)/(c+1)^14, 0, -212837625*c*(323*c^14-18768*c^13-327831*c^12
-1255664*c^11-2520045*c^10-3232416*c^9-2888127*c^8-1878624*c^7-901791*c^6-\
330000*c^5-71181*c^4-14256*c^3+1353*c^2+99)/(c+1)^15, 0, 638512875*c*(20349*c^
15+1093355*c^14+7803425*c^13+26283695*c^12+53240665*c^11+72912895*c^10+71651965
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165*c^2+495*c-495)/(c+1)^16], [0, 155925*c*(91*c^5+91*c^4+70*c^3+14*c^2+7*c-1)/
(c+1)^6, 0, 155925*c*(273*c^7+651*c^6+1029*c^5+623*c^4+379*c^3+41*c^2+15*c-3)/(
c+1)^8, 0, 155925*c*(693*c^9+5145*c^8+11340*c^7+14940*c^6+13270*c^5+5902*c^4+
2300*c^3+140*c^2+45*c-15)/(c+1)^10, 0, 467775*c*(273*c^11+9247*c^10+39831*c^9+
88425*c^8+113010*c^7+95662*c^6+54838*c^5+18634*c^4+5005*c^3+35*c^2+35*c-35)/(c+
1)^12, 0, -467775*c*(455*c^12-21736*c^11-298494*c^10-901728*c^9-1426515*c^8-\
1424880*c^7-961044*c^6-472416*c^5-124047*c^4-31080*c^3+3570*c^2+315)/(c+1)^13,
0, 2338875*c*(663*c^13-15977*c^12+311402*c^11+1244298*c^10+2370621*c^9+2725437*
c^8+2184732*c^7+1198428*c^6+537537*c^5+64449*c^4+19866*c^3-17094*c^2-693*c-693)
/(c+1)^14, 0, -91216125*c*(323*c^14-8466*c^13+181197*c^12+830428*c^11+1770747*c
^10+2313234*c^9+2096061*c^8+1356168*c^7+666897*c^6+221298*c^5+94479*c^4+15708*c
^3+9009*c^2+462*c+231)/(c+1)^15, 0, 638512875*c*(2261*c^15-143089*c^14-2765475*
c^13-11733425*c^12-26094815*c^11-37280925*c^10-37405335*c^9-27592125*c^8-\
15300945*c^7-6469155*c^6-2032569*c^5-570339*c^4-95205*c^3-26895*c^2-1485*c-495)
/(c+1)^16, 0], [-2027025*c*(17*c^4+20*c^3+14*c^2+4*c+1)/(c+1)^5, 0, -2027025*c*
(63*c^6+76*c^5+99*c^4+48*c^3+29*c^2+4*c+1)/(c+1)^7, 0, -2027025*c*(231*c^8+756*
c^7+1296*c^6+1060*c^5+846*c^4+284*c^3+120*c^2+12*c+3)/(c+1)^9, 0, -6081075*c*(
231*c^10+2100*c^9+5425*c^8+8400*c^7+8350*c^6+5144*c^5+2650*c^4+720*c^3+235*c^2+
20*c+5)/(c+1)^11, 0, -6081075*c*(315*c^12+12236*c^11+61238*c^10+152540*c^9+
223165*c^8+217240*c^7+145684*c^6+69272*c^5+26565*c^4+6300*c^3+1750*c^2+140*c+35
)/(c+1)^13, 0, 30405375*c*(119*c^13-6299*c^12-98286*c^11-336354*c^10-603051*c^9
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63)/(c+1)^14, 0, -91216125*c*(323*c^14-8466*c^13+181197*c^12+830428*c^11+
1770747*c^10+2313234*c^9+2096061*c^8+1356168*c^7+666897*c^6+221298*c^5+94479*c^
4+15708*c^3+9009*c^2+462*c+231)/(c+1)^15, 0, 638512875*c*(969*c^15-27455*c^14+
640101*c^13+3307317*c^12+7832045*c^11+11453013*c^10+11640057*c^9+8606169*c^8+
4807803*c^7+1996995*c^6+677391*c^5+80223*c^4+18447*c^3-12441*c^2-429*c-429)/(c+
1)^16, 0, -638512875*c*(52003*c^16-3560752*c^15-75426960*c^14-351391360*c^13-\
858663260*c^12-1353671280*c^11-1508398320*c^10-1246227840*c^9-783486990*c^8-\
380025360*c^7-142105392*c^6-41513472*c^5-7610460*c^4-1269840*c^3+102960*c^2+
6435)/(c+1)^17], [0, 2027025*c*(153*c^5+135*c^4+102*c^3+18*c^2+9*c-1)/(c+1)^6,
0, 2027025*c*(693*c^7+1149*c^6+1731*c^5+891*c^4+547*c^3+59*c^2+21*c-3)/(c+1)^8,
0, 2027025*c*(3003*c^9+12705*c^8+24300*c^7+25540*c^6+21730*c^5+8742*c^4+3500*c^
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255*c^9+961*c^8+1870*c^7+1918*c^6+1372*c^5+560*c^4+210*c^3+18*c^2+5*c-1)/(c+1)^
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82276176*c^15-603841896*c^14-1846054992*c^13-3006158106*c^12-3486026232*c^11-\
2689954228*c^10-1651493272*c^9-640576794*c^8-215014800*c^7-16147560*c^6-144144*
c^5+5516511*c^4+732732*c^3+306306*c^2+12012*c+3003)/(c+1)^21, 0, 86285437363125
*c*(561*c^21-14499*c^20+184350*c^19-1557370*c^18+10125885*c^17-58854951*c^16+
495183144*c^15+1965250440*c^14+4702714290*c^13+5790426570*c^12+5573884212*c^11+
2987045412*c^10+1341515890*c^9-43084470*c^8-141312600*c^7-188001528*c^6-\
52804323*c^5-23352615*c^4-2955810*c^3-810810*c^2-32175*c-6435)/(c+1)^22, 0], [
638512875*(9694845*c^14+60843510*c^13+179150335*c^12+327202876*c^11+413530117*c
^10+382167786*c^9+266128863*c^8+141772488*c^7+58032975*c^6+18156138*c^5+4273269
*c^4+732732*c^3+87087*c^2+6006*c+429)/(c+1)^14, 0, 638512875*(1002915*c^16+
25110020*c^15+146930762*c^14+445597880*c^13+864918418*c^12+1181592412*c^11+
1195892698*c^10+923551200*c^9+553238400*c^8+258901500*c^7+94588494*c^6+26738712
*c^5+5747742*c^4+900900*c^3+112398*c^2+6864*c+429)/(c+1)^16, 0, -638512875*(
185725*c^17-11284651*c^16-177392892*c^15-886331380*c^14-2429384048*c^13-\
4361161896*c^12-5579153580*c^11-5330151684*c^10-3907293390*c^9-2231464950*c^8-\
998686260*c^7-350233884*c^6-94822728*c^5-20420400*c^4-3042468*c^3-396396*c^2-\
21879*c-1287)/(c+1)^17, 0, 1915538625*(52003*c^18-1739678*c^17+53812769*c^16+
658113520*c^15+2889270820*c^14+7218771560*c^13+12033543340*c^12+14447633200*c^
11+13053398930*c^10+9093075420*c^9+4961051238*c^8+2120104272*c^7+727771044*c^6+
189189000*c^5+41141100*c^4+5800080*c^3+761475*c^2+38610*c+2145)/(c+1)^18, 0, -\
13408770375*(14535*c^19-411179*c^18+6741061*c^17-139359161*c^16-1408649780*c^15
-5523091100*c^14-12643549100*c^13-19643414180*c^12-22163943230*c^11-18990844730
*c^10-12552149610*c^9-6639481134*c^8-2724440004*c^7-927242316*c^6-231445500*c^5
-50287380*c^4-6694545*c^3-864435*c^2-40755*c-2145)/(c+1)^19, 0, 201131555625*(
3553*c^20-93568*c^19+1285740*c^18-13637748*c^17+207338403*c^16+1752361056*c^15+
6214983072*c^14+13055294928*c^13+19007363466*c^12+20099050400*c^11+16606968664*
c^10+10512617544*c^9+5468781318*c^8+2162997408*c^7+727391808*c^6+173638608*c^5+
37021413*c^4+4653792*c^3+581724*c^2+25740*c+1287)/(c+1)^20, 0, -6637341335625*(
663*c^21-16981*c^20+217722*c^19-1928766*c^18+14693763*c^17-169446537*c^16-\
1172465352*c^15-3834970728*c^14-7321645890*c^13-10295149722*c^12-10405031780*c^
11-8439207348*c^10-5148889746*c^9-2621273226*c^8-993347784*c^7-324313704*c^6-\
73585941*c^5-15123537*c^4-1797510*c^3-215358*c^2-9009*c-429)/(c+1)^21, 0, 
86285437363125*(495*c^22-12714*c^21+160447*c^20-1343300*c^19+8645385*c^18-\
49863450*c^17+429640449*c^16+2215753488*c^15+7464234390*c^14+13745950764*c^13+
18947228118*c^12+18284610344*c^11+14213209010*c^10+8207138940*c^9+3972036354*c^
8+1421101968*c^7+440459019*c^6+94444350*c^5+18453435*c^4+2078076*c^3+237237*c^2
+9438*c+429)/(c+1)^22, 0, -86285437363125*(7293*c^23-193413*c^22+2509929*c^21-\
21400225*c^20+136783675*c^19-722069523*c^18+3605284263*c^17-28351745679*c^16-\
134388830430*c^15-360246251250*c^14-566700748614*c^13-672255522666*c^12-\
576309776042*c^11-401417486470*c^10-211623404850*c^9-93998947614*c^8-\
31265579631*c^7-9032720697*c^6-1823256435*c^5-335660325*c^4-35900865*c^3-\
3893175*c^2-148005*c-6435)/(c+1)^23], [0, 638512875*c*(9694845*c^15+65189475*c^
14+206711925*c^13+409003595*c^12+563904705*c^11+573251679*c^10+443548105*c^9+
265823415*c^8+124356375*c^7+45390345*c^6+12819807*c^5+2747745*c^4+435435*c^3+
45045*c^2+6435*c-6435)/(c+1)^16, 0, -1915538625*c*(334305*c^16-17681020*c^15-\
116412430*c^14-351178520*c^13-659284990*c^12-863863364*c^11-836529694*c^10-\
618114640*c^9-354534180*c^8-159116100*c^7-55777722*c^6-15207192*c^5-3093090*c^4
-540540*c^3+81510*c^2+6435)/(c+1)^17, 0, 9577693125*c*(37145*c^17-1210927*c^16+
32859436*c^15+207933188*c^14+593624360*c^13+1054790464*c^12+1312655708*c^11+
1210504724*c^10+854755330*c^9+469175850*c^8+202551492*c^7+67759692*c^6+18546528
*c^5+2090088*c^4+386100*c^3-214500*c^2-6435*c-6435)/(c+1)^18, 0, -67043851875*c
*(7429*c^18-211242*c^17+3390531*c^16-62266512*c^15-370662564*c^14-997794984*c^
13-1673633052*c^12-1978730832*c^11-1736019714*c^10-1174433260*c^9-614097198*c^8
-260056368*c^7-72149220*c^6-18522504*c^5+653796*c^4+34320*c^3+353925*c^2+12870*
c+6435)/(c+1)^19, 0, 603394666875*c*(2261*c^19-60401*c^18+836927*c^17-8757099*c
^16+120521892*c^15+657263532*c^14+1667981436*c^13+2629576404*c^12+2963391366*c^
11+2459634034*c^10+1618228898*c^9+750314422*c^8+295510644*c^7+50963484*c^6+
10294284*c^5-5537532*c^4-727155*c^3-499785*c^2-19305*c-6435)/(c+1)^20, 0, -\
6637341335625*c*(969*c^20-25160*c^19+327200*c^18-2932660*c^17+22282815*c^16-\
239412864*c^15-1152452880*c^14-2780682480*c^13-4070162070*c^12-4451250960*c^11-\
3285058192*c^10-1983495800*c^9-688040210*c^8-216490560*c^7+23234640*c^6+
11510928*c^5+12966525*c^4+1698840*c^3+652080*c^2+25740*c+6435)/(c+1)^21, 0, 
86285437363125*c*(561*c^21-14499*c^20+184350*c^19-1557370*c^18+10125885*c^17-\
58854951*c^16+495183144*c^15+1965250440*c^14+4702714290*c^13+5790426570*c^12+
5573884212*c^11+2987045412*c^10+1341515890*c^9-43084470*c^8-141312600*c^7-\
188001528*c^6-52804323*c^5-23352615*c^4-2955810*c^3-810810*c^2-32175*c-6435)/(c
+1)^22, 0, -1294281560446875*c*(c-1)*(429*c^21-10857*c^20+134064*c^19-1083740*c
^18+6534535*c^17-32273199*c^16+148085952*c^15-1011521424*c^14-3439188270*c^13-\
7697465370*c^12-9646090272*c^11-9724786344*c^10-6726481762*c^9-4031817790*c^8-\
1684837440*c^7-654365712*c^6-162681519*c^5-42639597*c^4-5525520*c^3-1021020*c^2
-45045*c-6435)/(c+1)^23, 0], [638512875*(300540195*c^16+2326762800*c^15+
8464602600*c^14+19199507600*c^13+30381192660*c^12+35555183664*c^11+31827916120*
c^10+22227313680*c^9+12237040530*c^8+5328145680*c^7+1828538712*c^6+489368880*c^
5+100120020*c^4+15135120*c^3+1595880*c^2+102960*c+6435)/(c+1)^16, 0, -638512875
*(9694845*c^17-541239795*c^16-4176583800*c^15-14575995160*c^14-31535749700*c^13
-47592322596*c^12-53189560424*c^11-45548880520*c^10-30484950210*c^9-16112695170
*c^8-6746450568*c^7-2229979752*c^6-575554980*c^5-113813700*c^4-16525080*c^3-\
1904760*c^2-109395*c-6435)/(c+1)^17, 0, 1915538625*(1002915*c^18-34247690*c^17+
992625835*c^16+7514866320*c^15+25014625180*c^14+51484444648*c^13+74012932812*c^
12+78958772464*c^11+64688736970*c^10+41505930180*c^9+21074258634*c^8+8489800176
*c^7+2706555852*c^6+672071400*c^5+132947100*c^4+18429840*c^3+2220075*c^2+115830
*c+6435)/(c+1)^18, 0, -9577693125*(185725*c^19-5504889*c^18+93123807*c^17-\
1853630139*c^16-13533511980*c^15-42745774980*c^14-83492394804*c^13-114235554108
*c^12-116295338874*c^11-91181039950*c^10-56098541070*c^9-27395748666*c^8-\
10602867132*c^7-3311744436*c^6-792663300*c^5-158240940*c^4-20855835*c^3-2541825
*c^2-122265*c-6435)/(c+1)^19, 0, 67043851875*(52003*c^20-1440580*c^19+20878074*
c^18-232022052*c^17+3525940071*c^16+24349847472*c^15+72681912120*c^14+
134408562288*c^13+174978262134*c^12+169894514504*c^11+127628470684*c^10+
75172485960*c^9+35689840758*c^8+13321891440*c^7+4125329208*c^6+952002480*c^5+
190422375*c^4+23809500*c^3+2870010*c^2+128700*c+6435)/(c+1)^20, 0, -\
603394666875*(20349*c^21-544255*c^20+7344170*c^19-69083190*c^18+562782525*c^17-\
6843354519*c^16-43656353640*c^15-122944029480*c^14-214558300710*c^13-\
266164487550*c^12-245739660452*c^11-178790895140*c^10-101405986110*c^9-\
47380900710*c^8-17117803560*c^7-5254532712*c^6-1165693815*c^5-230224995*c^4-\
27297270*c^3-3204630*c^2-135135*c-6435)/(c+1)^21, 0, 6637341335625*(10659*c^22-\
281214*c^21+3670485*c^20-32106100*c^19+219350745*c^18-1380639654*c^17+
13621215999*c^16+77618543760*c^15+207257143950*c^14+338282757540*c^13+
406782285858*c^12+361058689992*c^11+258179594530*c^10+141859188900*c^9+
65242212750*c^8+22740685968*c^7+6835155327*c^6+1450281690*c^5+278388825*c^4+
31325580*c^3+3545685*c^2+141570*c+6435)/(c+1)^22, 0, -86285437363125*(7293*c^23
-193413*c^22+2509929*c^21-21400225*c^20+136783675*c^19-722069523*c^18+
3605284263*c^17-28351745679*c^16-134388830430*c^15-360246251250*c^14-\
566700748614*c^13-672255522666*c^12-576309776042*c^11-401417486470*c^10-\
211623404850*c^9-93998947614*c^8-31265579631*c^7-9032720697*c^6-1823256435*c^5-\
335660325*c^4-35900865*c^3-3893175*c^2-148005*c-6435)/(c+1)^23, 0, 
3882844681340625*(2145*c^24-58344*c^23+773652*c^22-6693144*c^21+42800450*c^20-\
218853880*c^19+962759364*c^18-4120324872*c^17+28351745679*c^16+119456738160*c^
15+288197001000*c^14+412145998992*c^13+448170348444*c^12+354652169872*c^11+
229381420840*c^10+112865815920*c^9+46999473807*c^8+14713213944*c^7+4014542532*c
^6+767686920*c^5+134264130*c^4+13676520*c^3+1415700*c^2+51480*c+2145)/(c+1)^24]
]]:
if N<=16 then
[seq([seq([seq(gu[i1+1][j1+1][k1+1],k1=0..N)],j1=0..N)],i1=0..N)]
else
 print(`For pre-computed staff we need N<=16,  so I am doing it from scratch`):
TavlaMom(N):
fi:

end:

TavlaMomPC:=proc(N) local gu,i1,j1,k1:
option remember:
gu:=
[[[1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0, 2027025], [0,
-1/2, 0, -3/2, 0, -15/2, 0, -105/2, 0, -945/2, 0, -10395/2, 0, -135135/2, 0, -\
2027025/2, 0], [1, 0, 3/2, 0, 6, 0, 75/2, 0, 315, 0, 6615/2, 0, 41580, 0, 
1216215/2, 0, 10135125], [0, -3/2, 0, -21/4, 0, -30, 0, -945/4, 0, -4725/2, 0,
-114345/4, 0, -405405, 0, -26351325/4, 0], [3, 0, 6, 0, 57/2, 0, 405/2, 0, 1890
, 0, 21735, 0, 592515/2, 0, 9324315/2, 0, 83108025], [0, -15/2, 0, -30, 0, -765
/4, 0, -6615/4, 0, -17955, 0, -467775/2, 0, -14189175/4, 0, -245270025/4, 0], [
15, 0, 75/2, 0, 405/2, 0, 6345/4, 0, 16065, 0, 198450, 0, 5769225/2, 0, 
192567375/4, 0, 906080175], [0, -105/2, 0, -945/4, 0, -6615/4, 0, -123795/8, 0,
-360045/2, 0, -2494800, 0, -160134975/4, 0, -5831750925/8, 0], [105, 0, 315, 0,
1890, 0, 16065, 0, 348705/2, 0, 4578525/2, 0, 70322175/2, 0, 1234458225/2, 0, 
12174312150], [0, -945/2, 0, -4725/2, 0, -17955, 0, -360045/2, 0, -8899065/4, 0
, -130197375/4, 0, -2195268075/4, 0, -41843877075/4, 0], [945, 0, 6615/2, 0, 
21735, 0, 198450, 0, 4578525/2, 0, 126908775/4, 0, 511745850, 0, 37587124575/4,
0, 193256563500], [0, -10395/2, 0, -114345/4, 0, -467775/2, 0, -2494800, 0, -\
130197375/4, 0, -3999944025/8, 0, -17629036425/2, 0, -1400684410125/8, 0], [
10395, 0, 41580, 0, 592515/2, 0, 5769225/2, 0, 70322175/2, 0, 511745850, 0, 
34516649475/4, 0, 660264880275/4, 0, 3526506608625], [0, -135135/2, 0, -405405,
0, -14189175/4, 0, -160134975/4, 0, -2195268075/4, 0, -17629036425/2, 0, -\
1294910191575/8, 0, -26712186125625/8, 0], [135135, 0, 1216215/2, 0, 9324315/2,
0, 192567375/4, 0, 1234458225/2, 0, 37587124575/4, 0, 660264880275/4, 0, 
26231258228175/8, 0, 72556772037750], [0, -2027025/2, 0, -26351325/4, 0, -\
245270025/4, 0, -5831750925/8, 0, -41843877075/4, 0, -1400684410125/8, 0, -\
26712186125625/8, 0, -1141410084940125/16, 0], [2027025, 0, 10135125, 0, 
83108025, 0, 906080175, 0, 12174312150, 0, 193256563500, 0, 3526506608625, 0, 
72556772037750, 0, 3318113246072625/2]], [[0, -1/2, 0, -3/2, 0, -15/2, 0, -105/
2, 0, -945/2, 0, -10395/2, 0, -135135/2, 0, -2027025/2, 0], [-1/2, 0, 0, 0, 3/2
, 0, 15, 0, 315/2, 0, 1890, 0, 51975/2, 0, 405405, 0, 14189175/2], [0, 0, 0, -3
/4, 0, -15/2, 0, -315/4, 0, -945, 0, -51975/4, 0, -405405/2, 0, -14189175/4, 0]
, [-3/2, 0, -3/4, 0, 3/2, 0, 135/4, 0, 945/2, 0, 27405/4, 0, 218295/2, 0, 
7702695/4, 0, 74999925/2], [0, 3/2, 0, 3/2, 0, -45/4, 0, -945/4, 0, -3780, 0, -\
62370, 0, -4459455/4, 0, -87162075/4, 0], [-15/2, 0, -15/2, 0, -45/4, 0, 135/2,
0, 1890, 0, 70875/2, 0, 2650725/4, 0, 26351325/2, 0, 565539975/2], [0, 15, 0, 
135/4, 0, 135/2, 0, -4725/8, 0, -36855/2, 0, -779625/2, 0, -8108100, 0, -\
1416890475/8, 0], [-105/2, 0, -315/4, 0, -945/4, 0, -4725/8, 0, 5670, 0, 411075
/2, 0, 19490625/4, 0, 893918025/8, 0, 5320940625/2], [0, 315/2, 0, 945/2, 0, 
1890, 0, 5670, 0, -257985/4, 0, -10446975/4, 0, -273648375/4, 0, -6853371525/4,
0], [-945/2, 0, -945, 0, -3780, 0, -36855/2, 0, -257985/4, 0, 822150, 0, 
148284675/4, 0, 1064188125, 0, 28903349475], [0, 1890, 0, 27405/4, 0, 70875/2,
0, 411075/2, 0, 822150, 0, -94022775/8, 0, -2329051725/4, 0, -145368097875/8, 0
], [-10395/2, 0, -51975/4, 0, -62370, 0, -779625/2, 0, -10446975/4, 0, -\
94022775/8, 0, 741423375/4, 0, 80154649575/8, 0, 337773310875], [0, 51975/2, 0,
218295/2, 0, 2650725/4, 0, 19490625/4, 0, 148284675/4, 0, 741423375/4, 0, -\
25619568975/8, 0, -1499866743375/8, 0], [-135135/2, 0, -405405/2, 0, -4459455/4
, 0, -8108100, 0, -273648375/4, 0, -2329051725/4, 0, -25619568975/8, 0, 
240463948725/4, 0, 7579786339125/2], [0, 405405, 0, 7702695/4, 0, 26351325/2, 0
, 893918025/8, 0, 1064188125, 0, 80154649575/8, 0, 240463948725/4, 0, -\
19498267663875/16, 0], [-2027025/2, 0, -14189175/4, 0, -87162075/4, 0, -\
1416890475/8, 0, -6853371525/4, 0, -145368097875/8, 0, -1499866743375/8, 0, -\
19498267663875/16, 0, 53058504373875/2], [0, 14189175/2, 0, 74999925/2, 0, 
565539975/2, 0, 5320940625/2, 0, 28903349475, 0, 337773310875, 0, 7579786339125
/2, 0, 53058504373875/2, 0]], [[1, 0, 3/2, 0, 6, 0, 75/2, 0, 315, 0, 6615/2, 0,
41580, 0, 1216215/2, 0, 10135125], [0, 0, 0, -3/4, 0, -15/2, 0, -315/4, 0, -945
, 0, -51975/4, 0, -405405/2, 0, -14189175/4, 0], [3/2, 0, 3/2, 0, 6, 0, 45, 0,
945/2, 0, 12285/2, 0, 93555, 0, 1621620, 0, 62837775/2], [0, -3/4, 0, -3, 0, -\
45/2, 0, -945/4, 0, -12285/4, 0, -93555/2, 0, -810810, 0, -62837775/4, 0], [6,
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34459425/4, 0, 36486450, 0, 918242325/4, 0, 7704722025/4, 0, 167162670675/8, 0,
1178056254375/4, 0, 5436298617750, 0, 257028888241125/2], [0, 26351325/2, 0, 
322296975/8, 0, 395269875/2, 0, 10429043625/8, 0, 41503336875/4, 0, 
1396427657625/16, 0, 994164546375/2, 0, -20222341264125/2, 0], [192567375/4, 0,
261486225/4, 0, 918242325/4, 0, 1246620375, 0, 73428980625/8, 0, 706833752625/8
, 0, 2269913270625/2, 0, 20222341264125, 0, 1972704044185875/4], [0, 893918025/
8, 0, 297972675, 0, 10429043625/8, 0, 31287130875/4, 0, 925205155875/16, 0, 
925205155875/2, 0, 10257709336875/4, 0, -211308812339625/4, 0], [1234458225/2,
0, 2596619025/4, 0, 7704722025/4, 0, 73428980625/8, 0, 120678933375/2, 0, 
522942044625, 0, 6154625602125, 0, 211308812339625/2, 0, 2650592092648500], [0,
1064188125, 0, 20645249625/8, 0, 41503336875/4, 0, 925205155875/16, 0, 
402263111250, 0, 6154625602125/2, 0, 16412334939000, 0, -1442233932764625/4, 0]
, [37587124575/4, 0, 16303362075/2, 0, 167162670675/8, 0, 706833752625/8, 0, 
522942044625, 0, 4103083734750, 0, 88216300297125/2, 0, 1442233932764625/2, 0,
36835434228718125/2], [0, 80154649575/8, 0, 45845224425/2, 0, 1396427657625/16,
0, 925205155875/2, 0, 6154625602125/2, 0, 22566960541125, 0, 428772250281375/4,
0, -6431583754220625/2, 0], [660264880275/4, 0, 1018938845925/8, 0, 
1178056254375/4, 0, 2269913270625/2, 0, 6154625602125, 0, 88216300297125/2, 0,
428772250281375, 0, 6431583754220625, 0, 160789593855515625], [0, 240463948725/
4, 0, 2143487721375/16, 0, 994164546375/2, 0, 10257709336875/4, 0, 
16412334939000, 0, 428772250281375/4, 0, 0, 0, -83610588804868125/2, 0], [
26231258228175/8, 0, 19498267663875/8, 0, 5436298617750, 0, 20222341264125, 0,
211308812339625/2, 0, 1442233932764625/2, 0, 6431583754220625, 0, 
83610588804868125, 0, 1839432953707098750], [0, -19498267663875/16, 0, -\
2718149308875, 0, -20222341264125/2, 0, -211308812339625/4, 0, -\
1442233932764625/4, 0, -6431583754220625/2, 0, -83610588804868125/2, 0, -\
919716476853549375, 0], [72556772037750, 0, 55776653682750, 0, 257028888241125/
2, 0, 1972704044185875/4, 0, 2650592092648500, 0, 36835434228718125/2, 0, 
160789593855515625, 0, 1839432953707098750, 0, 63460436902894906875/2]], [[0, -\
2027025/2, 0, -26351325/4, 0, -245270025/4, 0, -5831750925/8, 0, -41843877075/4
, 0, -1400684410125/8, 0, -26712186125625/8, 0, -1141410084940125/16, 0], [-\
2027025/2, 0, -14189175/4, 0, -87162075/4, 0, -1416890475/8, 0, -6853371525/4,
0, -145368097875/8, 0, -1499866743375/8, 0, -19498267663875/16, 0, 
53058504373875/2], [0, -14189175/4, 0, -62837775/4, 0, -845269425/8, 0, -\
3788509725/4, 0, -85858697925/8, 0, -1202319743625/8, 0, -41140023049125/16, 0,
-53058504373875, 0], [-26351325/4, 0, -62837775/4, 0, -614188575/8, 0, -\
2110133025/4, 0, -36054693675/8, 0, -350543568375/8, 0, -6855712738875/16, 0, -\
2718149308875, 0, 59201636744250], [0, -87162075/4, 0, -614188575/8, 0, -\
1708782075/4, 0, -25838487675/8, 0, -250169344425/8, 0, -6108652675125/16, 0, -\
11866761781875/2, 0, -118403273488500, 0], [-245270025/4, 0, -845269425/8, 0, -\
1708782075/4, 0, -20402006625/8, 0, -156435654375/8, 0, -2810095162875/16, 0, -\
1632038908500, 0, -20222341264125/2, 0, 880697615923125/4], [0, -1416890475/8,
0, -2110133025/4, 0, -20402006625/8, 0, -136003242375/8, 0, -2338872661125/16,
0, -1597559213250, 0, -91147074393375/4, 0, -880697615923125/2, 0], [-\
5831750925/8, 0, -3788509725/4, 0, -25838487675/8, 0, -136003242375/8, 0, -\
1890636622875/16, 0, -1971089245125/2, 0, -34876211745375/4, 0, -\
211308812339625/4, 0, 9160134437829375/8], [0, -6853371525/4, 0, -36054693675/8
, 0, -156435654375/8, 0, -1890636622875/16, 0, -925205155875, 0, -\
18463876806375/2, 0, -244133482217625/2, 0, -9160134437829375/4, 0], [-\
41843877075/4, 0, -85858697925/8, 0, -250169344425/8, 0, -2338872661125/16, 0,
-925205155875, 0, -14360793071625/2, 0, -121040970175125/2, 0, -\
1442233932764625/4, 0, 7600962618624375], [0, -145368097875/8, 0, -350543568375
/8, 0, -2810095162875/16, 0, -1971089245125/2, 0, -14360793071625/2, 0, -\
133350221379375/2, 0, -3313240115810625/4, 0, -15201925237248750, 0], [-\
1400684410125/8, 0, -1202319743625/8, 0, -6108652675125/16, 0, -1597559213250,
0, -18463876806375/2, 0, -133350221379375/2, 0, -2143861251406875/4, 0, -\
6431583754220625/2, 0, 237968598906163125/4], [0, -1499866743375/8, 0, -\
6855712738875/16, 0, -1632038908500, 0, -34876211745375/4, 0, -121040970175125/
2, 0, -2143861251406875/4, 0, -6431583754220625, 0, -237968598906163125/2, 0],
[-26712186125625/8, 0, -41140023049125/16, 0, -11866761781875/2, 0, -\
91147074393375/4, 0, -244133482217625/2, 0, -3313240115810625/4, 0, -\
6431583754220625, 0, -83610588804868125/2, 0, 919716476853549375/2], [0, -\
19498267663875/16, 0, -2718149308875, 0, -20222341264125/2, 0, -211308812339625
/4, 0, -1442233932764625/4, 0, -6431583754220625/2, 0, -83610588804868125/2, 0,
-919716476853549375, 0], [-1141410084940125/16, 0, -53058504373875, 0, -\
118403273488500, 0, -880697615923125/2, 0, -9160134437829375/4, 0, -\
15201925237248750, 0, -237968598906163125/2, 0, -919716476853549375, 0, 0], [0,
53058504373875/2, 0, 59201636744250, 0, 880697615923125/4, 0, 9160134437829375/
8, 0, 7600962618624375, 0, 237968598906163125/4, 0, 919716476853549375/2, 0, 0,
0]], [[2027025, 0, 10135125, 0, 83108025, 0, 906080175, 0, 12174312150, 0, 
193256563500, 0, 3526506608625, 0, 72556772037750, 0, 3318113246072625/2], [0,
14189175/2, 0, 74999925/2, 0, 565539975/2, 0, 5320940625/2, 0, 28903349475, 0,
337773310875, 0, 7579786339125/2, 0, 53058504373875/2, 0], [10135125, 0, 
62837775/2, 0, 182432250, 0, 2949321375/2, 0, 15239173950, 0, 194107914000, 0,
2999733486750, 0, 55776653682750, 0, 2469177176090625/2], [0, 74999925/2, 0, 
614188575/4, 0, 954728775, 0, 30946590675/4, 0, 75089114100, 0, 810272838375, 0
, 17303060399625/2, 0, 59201636744250, 0], [83108025, 0, 182432250, 0, 
1708782075/2, 0, 11560123575/2, 0, 50825624850, 0, 557421739875, 0, 
15130839598875/2, 0, 257028888241125/2, 0, 5513079152455875/2], [0, 565539975/2
, 0, 954728775, 0, 20402006625/4, 0, 146219448375/4, 0, 321810489000, 0, 
3229598121750, 0, 131591756921625/4, 0, 880697615923125/4, 0], [906080175, 0, 
2949321375/2, 0, 11560123575/2, 0, 136003242375/4, 0, 264344330250, 0, 
5166207671625/2, 0, 63011643069375/2, 0, 1972704044185875/4, 0, 
20519814834894375/2], [0, 5320940625/2, 0, 30946590675/4, 0, 146219448375/4, 0,
1890636622875/8, 0, 3821499556875/2, 0, 71803965358125/4, 0, 699575776774875/4,
0, 9160134437829375/8, 0], [12174312150, 0, 15239173950, 0, 50825624850, 0, 
264344330250, 0, 1850410311750, 0, 16412334939000, 0, 182587226196375, 0, 
2650592092648500, 0, 106998166092943125/2], [0, 28903349475, 0, 75089114100, 0,
321810489000, 0, 3821499556875/2, 0, 14360793071625, 0, 127195595777250, 0, 
2377737024287625/2, 0, 7600962618624375, 0], [193256563500, 0, 194107914000, 0,
557421739875, 0, 5166207671625/2, 0, 16412334939000, 0, 133350221379375, 0, 
1364275341804375, 0, 36835434228718125/2, 0, 719752691040508125/2], [0, 
337773310875, 0, 810272838375, 0, 3229598121750, 0, 71803965358125/4, 0, 
127195595777250, 0, 2143861251406875/2, 0, 19294751262661875/2, 0, 
237968598906163125/4, 0], [3526506608625, 0, 2999733486750, 0, 15130839598875/2
, 0, 63011643069375/2, 0, 182587226196375, 0, 1364275341804375, 0, 
12863167508441250, 0, 160789593855515625, 0, 6013530810196284375/2], [0, 
7579786339125/2, 0, 17303060399625/2, 0, 131591756921625/4, 0, 699575776774875/
4, 0, 2377737024287625/2, 0, 19294751262661875/2, 0, 83610588804868125, 0, 
919716476853549375/2, 0], [72556772037750, 0, 55776653682750, 0, 
257028888241125/2, 0, 1972704044185875/4, 0, 2650592092648500, 0, 
36835434228718125/2, 0, 160789593855515625, 0, 1839432953707098750, 0, 
63460436902894906875/2], [0, 53058504373875/2, 0, 59201636744250, 0, 
880697615923125/4, 0, 9160134437829375/8, 0, 7600962618624375, 0, 
237968598906163125/4, 0, 919716476853549375/2, 0, 0, 0], [3318113246072625/2, 0
, 2469177176090625/2, 0, 5513079152455875/2, 0, 20519814834894375/2, 0, 
106998166092943125/2, 0, 719752691040508125/2, 0, 6013530810196284375/2, 0, 
63460436902894906875/2, 0, 951906553543423603125/2]]]:



if N<=16 then
[seq([seq([seq(gu[i1+1][j1+1][k1+1],k1=0..N)],j1=0..N)],i1=0..N)]
else
 print(`For pre-computed staff we need N<=16,  so I am doing it from scratch`):
TavlaMom(N):
fi:

end:


#TavlaMforPC(N,n): a pre-computed version of TavlaMfor(N,n) for N<=5, otherwise it dies TavlaMfor(N,n).Try:
#TavlaMforPC(5,n);
TavlaMforPC:=proc(N,n) local i,j,k,gu:
option remember:
gu:=
[[[1, 0, 1/3*n^2*(2*n+1), 0, 1/15*n^3*(2*n+1)*(10*n^2-n-4), 0], [0, -1/3*n^3, 0
, -1/15*n^4*(10*n^2-n-4), 0, -1/63*n^4*(140*n^5-112*n^4-115*n^3+92*n^2+32*n-16)
], [1/3*n^2*(2*n+1), 0, 1/45*n^3*(30*n^3+14*n^2+3*n-2), 0, 1/315*n^3*(560*n^6-\
112*n^5-146*n^4-3*n^3-16*n^2+40*n-8), 0], [0, -1/15*n^4*(10*n^2-n-4), 0, -1/315
*n^3*(490*n^6-210*n^5-357*n^4+42*n^3+144*n^2-12*n+8), 0, -1/945*n^3*(5600*n^9-\
8120*n^8-2406*n^7+6237*n^6+3060*n^5-3336*n^4-1664*n^3+1280*n^2-432*n+96)], [1/
15*n^3*(2*n+1)*(10*n^2-n-4), 0, 1/315*n^3*(560*n^6-112*n^5-146*n^4-3*n^3-16*n^2
+40*n-8), 0, 1/4725*n^3*(26600*n^9-31920*n^8-2568*n^7+26472*n^6-7899*n^5-6480*n
^4+3928*n^3-5232*n^2+2304*n-480), 0], [0, -1/63*n^4*(140*n^5-112*n^4-115*n^3+92
*n^2+32*n-16), 0, -1/945*n^3*(5600*n^9-8120*n^8-2406*n^7+6237*n^6+3060*n^5-3336
*n^4-1664*n^3+1280*n^2-432*n+96), 0, -1/6237*n^3*(157080*n^12-437360*n^11+
261624*n^10+333608*n^9-301281*n^8-162960*n^7-47736*n^6+254240*n^5+44928*n^4-\
183584*n^3+119808*n^2-44352*n+8064)]], [[0, -1/3*n^3, 0, -1/15*n^4*(10*n^2-n-4)
, 0, -1/63*n^4*(140*n^5-112*n^4-115*n^3+92*n^2+32*n-16)], [-1/3*n^3, 0, -1/45*n
^3*(11*n^2+2*n+2), 0, 1/315*n^3*(140*n^6-420*n^5-13*n^4+144*n^3+12*n^2+40*n-8),
0], [0, -1/45*n^3*(11*n^2+2*n+2), 0, -1/315*n^3*(70*n^6+28*n^5+45*n^4-52*n^3+34
*n^2-28*n+8), 0, -1/945*n^3*(1400*n^9-2800*n^8+2588*n^7-861*n^6+226*n^5-102*n^4
-760*n^3+1024*n^2-496*n+96)], [-1/15*n^4*(10*n^2-n-4), 0, -1/315*n^3*(70*n^6+28
*n^5+45*n^4-52*n^3+34*n^2-28*n+8), 0, 1/4725*n^3*(1400*n^9-7560*n^8-1350*n^7+
6507*n^6+2472*n^5-4908*n^4+4768*n^3-4680*n^2+2256*n-480), 0], [0, 1/315*n^3*(
140*n^6-420*n^5-13*n^4+144*n^3+12*n^2+40*n-8), 0, 1/4725*n^3*(1400*n^9-7560*n^8
-1350*n^7+6507*n^6+2472*n^5-4908*n^4+4768*n^3-4680*n^2+2256*n-480), 0, -1/31185
*n^3*(46200*n^12-141680*n^11+330352*n^10-151580*n^9-423201*n^8+479004*n^7-\
127692*n^6-171016*n^5+574328*n^4-774688*n^3+552960*n^2-222912*n+40320)], [-1/63
*n^4*(140*n^5-112*n^4-115*n^3+92*n^2+32*n-16), 0, -1/945*n^3*(1400*n^9-2800*n^8
+2588*n^7-861*n^6+226*n^5-102*n^4-760*n^3+1024*n^2-496*n+96), 0, -1/31185*n^3*(
46200*n^12-141680*n^11+330352*n^10-151580*n^9-423201*n^8+479004*n^7-127692*n^6-\
171016*n^5+574328*n^4-774688*n^3+552960*n^2-222912*n+40320), 0]], [[1/3*n^2*(2*
n+1), 0, 1/45*n^3*(30*n^3+14*n^2+3*n-2), 0, 1/315*n^3*(560*n^6-112*n^5-146*n^4-\
3*n^3-16*n^2+40*n-8), 0], [0, -1/45*n^3*(11*n^2+2*n+2), 0, -1/315*n^3*(70*n^6+
28*n^5+45*n^4-52*n^3+34*n^2-28*n+8), 0, -1/945*n^3*(1400*n^9-2800*n^8+2588*n^7-\
861*n^6+226*n^5-102*n^4-760*n^3+1024*n^2-496*n+96)], [1/45*n^3*(30*n^3+14*n^2+3
*n-2), 0, 1/315*n^3*(140*n^6+294*n^5-72*n^4-9*n^3-54*n^2+24*n-8), 0, 1/4725*n^3
*(5600*n^9+8120*n^8-8686*n^7-5874*n^6+6475*n^5-3390*n^4+5216*n^3-4496*n^2+2240*
n-480), 0], [0, -1/315*n^3*(70*n^6+28*n^5+45*n^4-52*n^3+34*n^2-28*n+8), 0, -1/
4725*n^3*(2800*n^9+3290*n^8-9918*n^7+5757*n^6-3516*n^5+6594*n^4-5776*n^3+4072*n
^2-2208*n+480), 0, -1/31185*n^3*(92400*n^12-33880*n^11-435402*n^10+626956*n^9-\
90681*n^8-204888*n^7+191454*n^6-465956*n^5+727464*n^4-744976*n^3+529920*n^2-\
222336*n+40320)], [1/315*n^3*(560*n^6-112*n^5-146*n^4-3*n^3-16*n^2+40*n-8), 0,
1/4725*n^3*(5600*n^9+8120*n^8-8686*n^7-5874*n^6+6475*n^5-3390*n^4+5216*n^3-4496
*n^2+2240*n-480), 0, 1/155925*n^3*(462000*n^12+773080*n^11-2399408*n^10+1100176
*n^9+447582*n^8+1099791*n^7-2867640*n^6+3347384*n^5-3910048*n^4+3818240*n^3-\
2599968*n^2+1086336*n-201600), 0], [0, -1/945*n^3*(1400*n^9-2800*n^8+2588*n^7-\
861*n^6+226*n^5-102*n^4-760*n^3+1024*n^2-496*n+96), 0, -1/31185*n^3*(92400*n^12
-33880*n^11-435402*n^10+626956*n^9-90681*n^8-204888*n^7+191454*n^6-465956*n^5+
727464*n^4-744976*n^3+529920*n^2-222336*n+40320), 0, -1/2837835*n^3*(33633600*n
^15-12612600*n^14-364283920*n^13+823446624*n^12-371158944*n^11-572235067*n^10+
516766182*n^9+110459702*n^8-578801744*n^7+1443895728*n^6-2642784288*n^5+
3369542272*n^4-3039561984*n^3+1848398464*n^2-675202560*n+111444480)]], [[0, -1/
15*n^4*(10*n^2-n-4), 0, -1/315*n^3*(490*n^6-210*n^5-357*n^4+42*n^3+144*n^2-12*n
+8), 0, -1/945*n^3*(5600*n^9-8120*n^8-2406*n^7+6237*n^6+3060*n^5-3336*n^4-1664*
n^3+1280*n^2-432*n+96)], [-1/15*n^4*(10*n^2-n-4), 0, -1/315*n^3*(70*n^6+28*n^5+
45*n^4-52*n^3+34*n^2-28*n+8), 0, 1/4725*n^3*(1400*n^9-7560*n^8-1350*n^7+6507*n^
6+2472*n^5-4908*n^4+4768*n^3-4680*n^2+2256*n-480), 0], [0, -1/315*n^3*(70*n^6+
28*n^5+45*n^4-52*n^3+34*n^2-28*n+8), 0, -1/4725*n^3*(2800*n^9+3290*n^8-9918*n^7
+5757*n^6-3516*n^5+6594*n^4-5776*n^3+4072*n^2-2208*n+480), 0, -1/31185*n^3*(
92400*n^12-33880*n^11-435402*n^10+626956*n^9-90681*n^8-204888*n^7+191454*n^6-\
465956*n^5+727464*n^4-744976*n^3+529920*n^2-222336*n+40320)], [-1/315*n^3*(490*
n^6-210*n^5-357*n^4+42*n^3+144*n^2-12*n+8), 0, -1/4725*n^3*(2800*n^9+3290*n^8-\
9918*n^7+5757*n^6-3516*n^5+6594*n^4-5776*n^3+4072*n^2-2208*n+480), 0, -1/51975*
n^3*(141680*n^11+80014*n^10-716342*n^9+899763*n^8-842358*n^7+1120788*n^6-\
1397660*n^5+1433408*n^4-1258768*n^3+849024*n^2-359424*n+67200), 0], [0, 1/4725*
n^3*(1400*n^9-7560*n^8-1350*n^7+6507*n^6+2472*n^5-4908*n^4+4768*n^3-4680*n^2+
2256*n-480), 0, -1/51975*n^3*(141680*n^11+80014*n^10-716342*n^9+899763*n^8-\
842358*n^7+1120788*n^6-1397660*n^5+1433408*n^4-1258768*n^3+849024*n^2-359424*n+
67200), 0, -1/14189175*n^3*(42042000*n^15+29429400*n^14-288648360*n^13+
454726272*n^12-616876650*n^11+1522875611*n^10-3466829696*n^9+6088185732*n^8-\
9388493136*n^7+13388708088*n^6-16915309920*n^5+17936178880*n^4-15034561408*n^3+
9035218560*n^2-3339138048*n+557222400)], [-1/945*n^3*(5600*n^9-8120*n^8-2406*n^
7+6237*n^6+3060*n^5-3336*n^4-1664*n^3+1280*n^2-432*n+96), 0, -1/31185*n^3*(
92400*n^12-33880*n^11-435402*n^10+626956*n^9-90681*n^8-204888*n^7+191454*n^6-\
465956*n^5+727464*n^4-744976*n^3+529920*n^2-222336*n+40320), 0, -1/14189175*n^3
*(42042000*n^15+29429400*n^14-288648360*n^13+454726272*n^12-616876650*n^11+
1522875611*n^10-3466829696*n^9+6088185732*n^8-9388493136*n^7+13388708088*n^6-\
16915309920*n^5+17936178880*n^4-15034561408*n^3+9035218560*n^2-3339138048*n+
557222400), 0]], [[1/15*n^3*(2*n+1)*(10*n^2-n-4), 0, 1/315*n^3*(560*n^6-112*n^5
-146*n^4-3*n^3-16*n^2+40*n-8), 0, 1/4725*n^3*(26600*n^9-31920*n^8-2568*n^7+
26472*n^6-7899*n^5-6480*n^4+3928*n^3-5232*n^2+2304*n-480), 0], [0, 1/315*n^3*(
140*n^6-420*n^5-13*n^4+144*n^3+12*n^2+40*n-8), 0, 1/4725*n^3*(1400*n^9-7560*n^8
-1350*n^7+6507*n^6+2472*n^5-4908*n^4+4768*n^3-4680*n^2+2256*n-480), 0, -1/31185
*n^3*(46200*n^12-141680*n^11+330352*n^10-151580*n^9-423201*n^8+479004*n^7-\
127692*n^6-171016*n^5+574328*n^4-774688*n^3+552960*n^2-222912*n+40320)], [1/315
*n^3*(560*n^6-112*n^5-146*n^4-3*n^3-16*n^2+40*n-8), 0, 1/4725*n^3*(5600*n^9+
8120*n^8-8686*n^7-5874*n^6+6475*n^5-3390*n^4+5216*n^3-4496*n^2+2240*n-480), 0,
1/155925*n^3*(462000*n^12+773080*n^11-2399408*n^10+1100176*n^9+447582*n^8+
1099791*n^7-2867640*n^6+3347384*n^5-3910048*n^4+3818240*n^3-2599968*n^2+1086336
*n-201600), 0], [0, 1/4725*n^3*(1400*n^9-7560*n^8-1350*n^7+6507*n^6+2472*n^5-\
4908*n^4+4768*n^3-4680*n^2+2256*n-480), 0, -1/51975*n^3*(141680*n^11+80014*n^10
-716342*n^9+899763*n^8-842358*n^7+1120788*n^6-1397660*n^5+1433408*n^4-1258768*n
^3+849024*n^2-359424*n+67200), 0, -1/14189175*n^3*(42042000*n^15+29429400*n^14-\
288648360*n^13+454726272*n^12-616876650*n^11+1522875611*n^10-3466829696*n^9+
6088185732*n^8-9388493136*n^7+13388708088*n^6-16915309920*n^5+17936178880*n^4-\
15034561408*n^3+9035218560*n^2-3339138048*n+557222400)], [1/4725*n^3*(26600*n^9
-31920*n^8-2568*n^7+26472*n^6-7899*n^5-6480*n^4+3928*n^3-5232*n^2+2304*n-480),
0, 1/155925*n^3*(462000*n^12+773080*n^11-2399408*n^10+1100176*n^9+447582*n^8+
1099791*n^7-2867640*n^6+3347384*n^5-3910048*n^4+3818240*n^3-2599968*n^2+1086336
*n-201600), 0, 1/23648625*n^3*(140140000*n^15+504504000*n^14-1499618120*n^13-\
1096791696*n^12+7342193508*n^11-10641870640*n^10+11263147417*n^9-15566404476*n^
8+22207220072*n^7-27552084528*n^6+31164782784*n^5-30989982368*n^4+25030908160*n
^3-14872399872*n^2+5518608384*n-928704000), 0], [0, -1/31185*n^3*(46200*n^12-\
141680*n^11+330352*n^10-151580*n^9-423201*n^8+479004*n^7-127692*n^6-171016*n^5+
574328*n^4-774688*n^3+552960*n^2-222912*n+40320), 0, -1/14189175*n^3*(42042000*
n^15+29429400*n^14-288648360*n^13+454726272*n^12-616876650*n^11+1522875611*n^10
-3466829696*n^9+6088185732*n^8-9388493136*n^7+13388708088*n^6-16915309920*n^5+
17936178880*n^4-15034561408*n^3+9035218560*n^2-3339138048*n+557222400), 0, -1/
2837835*n^3*(39239200*n^18+66146080*n^17-816055240*n^16+1114633520*n^15+
3208398492*n^14-13589761044*n^13+25028291837*n^12-38043392560*n^11+62580129596*
n^10-103184180072*n^9+157753326632*n^8-224678523360*n^7+293133737664*n^6-\
336053442624*n^5+322828696448*n^4-243844376832*n^3+132045454336*n^2-44452356096
*n+6864979968)]], [[0, -1/63*n^4*(140*n^5-112*n^4-115*n^3+92*n^2+32*n-16), 0, -\
1/945*n^3*(5600*n^9-8120*n^8-2406*n^7+6237*n^6+3060*n^5-3336*n^4-1664*n^3+1280*
n^2-432*n+96), 0, -1/6237*n^3*(157080*n^12-437360*n^11+261624*n^10+333608*n^9-\
301281*n^8-162960*n^7-47736*n^6+254240*n^5+44928*n^4-183584*n^3+119808*n^2-\
44352*n+8064)], [-1/63*n^4*(140*n^5-112*n^4-115*n^3+92*n^2+32*n-16), 0, -1/945*
n^3*(1400*n^9-2800*n^8+2588*n^7-861*n^6+226*n^5-102*n^4-760*n^3+1024*n^2-496*n+
96), 0, -1/31185*n^3*(46200*n^12-141680*n^11+330352*n^10-151580*n^9-423201*n^8+
479004*n^7-127692*n^6-171016*n^5+574328*n^4-774688*n^3+552960*n^2-222912*n+
40320), 0], [0, -1/945*n^3*(1400*n^9-2800*n^8+2588*n^7-861*n^6+226*n^5-102*n^4-\
760*n^3+1024*n^2-496*n+96), 0, -1/31185*n^3*(92400*n^12-33880*n^11-435402*n^10+
626956*n^9-90681*n^8-204888*n^7+191454*n^6-465956*n^5+727464*n^4-744976*n^3+
529920*n^2-222336*n+40320), 0, -1/2837835*n^3*(33633600*n^15-12612600*n^14-\
364283920*n^13+823446624*n^12-371158944*n^11-572235067*n^10+516766182*n^9+
110459702*n^8-578801744*n^7+1443895728*n^6-2642784288*n^5+3369542272*n^4-\
3039561984*n^3+1848398464*n^2-675202560*n+111444480)], [-1/945*n^3*(5600*n^9-\
8120*n^8-2406*n^7+6237*n^6+3060*n^5-3336*n^4-1664*n^3+1280*n^2-432*n+96), 0, -1
/31185*n^3*(92400*n^12-33880*n^11-435402*n^10+626956*n^9-90681*n^8-204888*n^7+
191454*n^6-465956*n^5+727464*n^4-744976*n^3+529920*n^2-222336*n+40320), 0, -1/
14189175*n^3*(42042000*n^15+29429400*n^14-288648360*n^13+454726272*n^12-\
616876650*n^11+1522875611*n^10-3466829696*n^9+6088185732*n^8-9388493136*n^7+
13388708088*n^6-16915309920*n^5+17936178880*n^4-15034561408*n^3+9035218560*n^2-\
3339138048*n+557222400), 0], [0, -1/31185*n^3*(46200*n^12-141680*n^11+330352*n^
10-151580*n^9-423201*n^8+479004*n^7-127692*n^6-171016*n^5+574328*n^4-774688*n^3
+552960*n^2-222912*n+40320), 0, -1/14189175*n^3*(42042000*n^15+29429400*n^14-\
288648360*n^13+454726272*n^12-616876650*n^11+1522875611*n^10-3466829696*n^9+
6088185732*n^8-9388493136*n^7+13388708088*n^6-16915309920*n^5+17936178880*n^4-\
15034561408*n^3+9035218560*n^2-3339138048*n+557222400), 0, -1/2837835*n^3*(
39239200*n^18+66146080*n^17-816055240*n^16+1114633520*n^15+3208398492*n^14-\
13589761044*n^13+25028291837*n^12-38043392560*n^11+62580129596*n^10-\
103184180072*n^9+157753326632*n^8-224678523360*n^7+293133737664*n^6-\
336053442624*n^5+322828696448*n^4-243844376832*n^3+132045454336*n^2-44452356096
*n+6864979968)], [-1/6237*n^3*(157080*n^12-437360*n^11+261624*n^10+333608*n^9-\
301281*n^8-162960*n^7-47736*n^6+254240*n^5+44928*n^4-183584*n^3+119808*n^2-\
44352*n+8064), 0, -1/2837835*n^3*(33633600*n^15-12612600*n^14-364283920*n^13+
823446624*n^12-371158944*n^11-572235067*n^10+516766182*n^9+110459702*n^8-\
578801744*n^7+1443895728*n^6-2642784288*n^5+3369542272*n^4-3039561984*n^3+
1848398464*n^2-675202560*n+111444480), 0, -1/2837835*n^3*(39239200*n^18+
66146080*n^17-816055240*n^16+1114633520*n^15+3208398492*n^14-13589761044*n^13+
25028291837*n^12-38043392560*n^11+62580129596*n^10-103184180072*n^9+
157753326632*n^8-224678523360*n^7+293133737664*n^6-336053442624*n^5+
322828696448*n^4-243844376832*n^3+132045454336*n^2-44452356096*n+6864979968), 0
]]]:

if N<=5 then
 [seq([seq([seq(gu[i+1][j+1][k+1],k=0..N)],j=0..N)],i=0..N)]:
else
TavlaMfor(N,n):
fi:

end:


#########end pre-computed
###Start GuessRat
#GuessRat1(gu,n,d): given a list gu 
#guesses a rational function R(n) of degree <=d , such that R(gu[i][1])=gu[i][2] for all i
#For example, try: GuessRat1([seq([i,i/(i+1)],i=1..20)],n,1);
GuessRat1:=proc(gu,n,d) local eq,var,a,i,b,R:

if nops(gu)=2*d+2+3 then
 print(`make list bigger`):
 FAIL:
fi:

var:={seq(a[i],i=0..d-1), seq(b[i],i=0..d)}:
R:=add(a[i]*n^i,i=0..d)/add(b[i]*n^i,i=0..d):
R:=subs(a[d]=1,R):
eq:={ seq(subs(n=gu[i][1],R)-gu[i][2],i=1..nops(gu))}:

var:=solve(eq,var):

if var=NULL then
 RETURN(FAIL):
fi:

normal(subs(var,R)):

end:

#GuessRat1g(gu,n,d1,d2): given a list gu 
#guesses a rational function R(n) of degree <=d , such that R(gu[i][1])=gu[i][2] for all i
#For example, try: GuessRat1g([seq([i,i/(i+1)],i=1..20)],n,1);
GuessRat1g:=proc(gu,n,d1,d2) local eq,var,a,i,b,R:

if nops(gu)=d1+d2+2+3 then
 print(`make list bigger`):
 FAIL:
fi:

var:={seq(a[i],i=0..d1-1), seq(b[i],i=0..d2)}:
R:=add(a[i]*n^i,i=0..d1)/add(b[i]*n^i,i=0..d2):
R:=subs(a[d1]=1,R):
eq:={ seq(subs(n=gu[i][1],R)-gu[i][2],i=1..nops(gu))}:

var:=solve(eq,var):

if var=NULL then
 RETURN(FAIL):
fi:

normal(subs(var,R)):

end:


#GuessRat(gu,n): given a list gu 
#guesses a rational function R(n) of degree <=d , such that R(gu[i][1])=gu[i][2] for all i
#For example, try: GuessRat([seq([i,i/(i+1)],i=1..20)],n);
GuessRat:=proc(gu,n) local R,d:

if {seq(gu[i][2],i=1..nops(gu))}={0} then
 RETURN(0):
fi:

for d from 1 to (nops(gu)-5)/2 do
 R:=GuessRat1(gu,n,d):
 if R<>FAIL then
  RETURN(factor(R)):
 fi:
od:
FAIL:
end:

#GuessRatg(gu,n): given a list gu 
#guesses a rational function R(n) of degree <=d , such that R(gu[i][1])=gu[i][2] for all i
#For example, try: GuessRatg([seq([i,i/(i+1)],i=1..20)],n);
GuessRatg:=proc(gu,n) local R,d1,d2:

if {seq(gu[i][2],i=1..nops(gu))}={0} then
 RETURN(0):
fi:

for d1 from 0 to (nops(gu)-5)/2 do
 for d2 from 0 to (nops(gu)-5)/2-d1 do
 R:=GuessRat1g(gu,n,d1,d2):
 if R<>FAIL then
  RETURN(factor(R)):
 fi:
od:
od:

FAIL:
end:

#GuessPol1g(gu,n,d): given a list gu 
#guesses a polynomial P(n) of degree <=d , such that P(gu[i][1])=gu[i][2] for all i
#For example, try: GuessRat1g([seq([i,i/(i+1)],i=1..20)],n,1);
GuessPol1g:=proc(gu,n,d) local eq,var,a,i,R:

if nops(gu)<d+2 then
 print(`make list bigger`):
 RETURN(FAIL):
fi:

var:={seq(a[i],i=0..d)}:
R:=add(a[i]*n^i,i=0..d):
eq:={ seq(subs(n=gu[i][1],R)-gu[i][2],i=1..nops(gu))}:

var:=solve(eq,var):

if var=NULL then
 RETURN(FAIL):
fi:

factor(normal(subs(var,R))):

end:

###end GuessRat
#GFq(a,b,c,q): the probability Generating function according to q[1]^(#12-#21)*q[2]^(#23-#32)*q[3]^(#31-#13)
#For all walks with unit steps from the origin to (a,b,c)
#Try
#GFq(3,3,3,q);
GFq:=proc(a,b,c,q) option remember:

if a<0 or b<0 or c<0 then
 RETURN(0):
fi:

if a=0 and b=0 and c=0 then
  RETURN(1):
fi:

expand(
a/(a+b+c)*q[1]^(-b)*q[3]^c*GFq(a-1,b,c,q)+
b/(a+b+c)*q[1]^a*q[2]^(-c)*GFq(a,b-1,c,q)+
c/(a+b+c)*q[3]^(-a)*q[2]^b*GFq(a,b,c-1,q)
):
end:


#Trun(F,var,R): given a function F in the list of variables var, finds its Taylor expansion to order R. Try
#Trun(1/(1-x-y),[x,y],2):
Trun:=proc(F,var,R) local gu,x,i,var1:

x:=var[nops(var)]:

if nops(var)=1 then
gu:=taylor(F,x=0,R+1):
gu:=add(coeff(gu,x,i)*x^i,i=0..R):
RETURN(gu):
fi:

var1:=[op(1..nops(var)-1,var)]:
gu:=taylor(F,x=0,R+1):

expand(add(Trun(coeff(gu,x,i),var1,R)*x^i,i=0..R)):

end:

#BM(a,b,c,t,R): the binomial moment generating function up to order R
#the coefficient of t[1]^i1*t[2]^i2*t[3]^i3 (for 0<=i1,i2,i3<=R) is
#the expectation of binomial(X1,i1)*binomial(X2,i2)*binomial(X3,i3).
#here X1 is the r.v. #of 12-#21, X2 is the r.v #23-32 , X3 is r.v. #31-#13
#Try
#BM(3,3,3,t,2);
BM:=proc(a,b,c,t,R) local gu:
option remember:

if a<0 or b<0 or c<0 then
 RETURN(0):
fi:

if a=0 and b=0 and c=0 then
  RETURN(1):
fi:

gu:=
a/(a+b+c)*(1+t[1])^(-b)*(1+t[3])^c*BM(a-1,b,c,t,R)+
b/(a+b+c)*(1+t[1])^a*(1+t[2])^(-c)*BM(a,b-1,c,t,R)+
c/(a+b+c)*(1+t[3])^(-a)*(1+t[2])^b*BM(a,b,c-1,t,R):

gu:=Trun(gu,[t[1],t[2],t[3]],R):
end:




#BMdata(i,j,k,R): collects data for the  binomial moments (i,j,k) for  GFq(n,n,n,q1,q2,q3) from n=1 to n=R. Try
#BMdata(1,1,1,10);
BMdata:=proc(i,j,k,R) local n1,t:
[seq(coeff(coeff(coeff(BM(n1,n1,n1,t,max(i,j,k)),t[1],i),t[2],j),t[3],k),n1=1..R)]:
end:

#GP1(L,n,d): inputs a list L of numbers and guesses a polynomial of degree d. Try:
#GP1([seq(i^2,i=1..10)],n,2);
GP1:=proc(L,n,d) local eq,var,a,i,P:

if nops(L)<d+3 then
 print(`the list must be at least of length`, d+3):
 RETURN(FAIL):
fi:

P:=add(a[i]*n^i,i=0..d):
var:={seq(a[i],i=0..d)}:
eq:={seq(subs(n=i,P)-L[i],i=1..d+2)}:

var:=solve(eq,var):

if var=NULL then
 RETURN(FAIL):
fi:

P:=subs(var,P):

if {seq(subs(n=i,P)-L[i],i=d+3..nops(L))}<>{0} then
 print(P, `did not work out`):
 RETURN(FAIL):
fi:

P:

end:


#GP(L,n): inputs a list L of numbers and guesses a polynomial of degree <=nops(L)-3, or returns FAIL. Try:
#GP([seq(i^2,i=1..10)],n);
GP:=proc(L,n) local d,gu:

for d from 0 to nops(L)-3 do
 gu:=GP1(L,n,d):
 if gu<>FAIL then
   RETURN(gu):
 fi:
od:
FAIL:
end:



#FMfor(n,i,j,k): the formula for the (i,j,k) mixed factorial moment for our statistics. Try
#FMfor(n,2,1,1);
FMfor:=proc(n,i,j,k) local gu:
option remember:

gu:=GP(BMdata(i,j,k,trunc((i+j+k)*3/2)+3),n) :

if gu=FAIL then
  RETURN(FAIL):
fi:

factor(gu*i!*j!*k!):

end:


#Mfor(n,i,j,k): the formula for the (i,j,k) mixed moment for our statistics. Try
#Mfor(n,2,1,1);
Mfor:=proc(n,i,j,k) local i1,j1,k1,mu:
option remember:

mu:=[i,j,k]:
if sort(mu)<>mu then
 RETURN(Mfor(n,op(sort(mu)) )):
fi:

factor(add(add(add(stirling2(i,i1)*stirling2(j,j1)*stirling2(k,k1)*FMfor(n,i1,j1,k1),i1=0..i),j1=0..j),k1=0..k)):
end:

#LimSM(i,j,k): the limit, as n goes to infinite of the (i,j,k) scaled moment. Try
#LimSM(2,2,2);
LimSM:=proc(i,j,k) local n,gu,lu,ku,c:

lu:=Mfor(n,2,0,0):
gu:=Mfor(n,i,j,k):

if lcoeff(gu,n)<0 then
 c:=-1:
elif lcoeff(gu,n)>0 then
 c:=1:
fi:

ku:=normal(gu^2/lu^(i+j+k)):

if degree(numer(ku),n)<degree(denom(ku),n) then
  RETURN(0):
elif degree(numer(ku),n)>degree(denom(ku),n) then
 RETURN(FAIL):
else
 c*sqrt(lcoeff(numer(ku),n)/lcoeff(denom(ku),n)):
fi:
  
end:

#LimSMpc(i,j,k): Like LimSM(i,j,k), but much faster ,for 0<=i,j,k<=5, since it uses pre-computed formulas for the moments. Try
#LimSMpc(2,2,2);
LimSMpc:=proc(i,j,k) local n,gu,ku,N,lu,c:

N:=max(5,i,j,k):

ku:=TavlaMforPC(N,n):

lu:=ku[3][1][1]:


gu:=ku[i+1][j+1][k+1]:

if lcoeff(gu,n)<0 then
 c:=-1:
elif lcoeff(gu,n)>0 then
 c:=1:
fi:

ku:=normal(gu^2/lu^(i+j+k)):

if degree(numer(ku),n)<degree(denom(ku),n) then
  RETURN(0):
elif degree(numer(ku),n)>degree(denom(ku),n) then
 RETURN(FAIL):
else
 c*sqrt(lcoeff(numer(ku),n)/lcoeff(denom(ku),n)):
fi:
  
end:


#Mijk(c,i,j,k): the (i,j,k) moment of exp(-x^2/2-y^2/-z^2/2 -(x*y+x*z+y*z)*c) normalized.
Mijk:=proc(c,i,j,k) local lu,a,b,x,y,z:
lu:=exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c):
a:=int(int(int(lu,x=-infinity..infinity),y=-infinity..infinity),z=-infinity..infinity):

b:=int(int(int(x^i*y^j*z^k*lu,x=-infinity..infinity),y=-infinity..infinity),z=-infinity..infinity):
b/a:
end:


#Sijk(c,i,j,k): the (i,j,k) standardized moment of exp(-x^2/2-y^2/-z^2/2 -(x*y+x*z+y*z)*c) normalized.
Sijk:=proc(c,i,j,k) local lu:
Mijk(c,i,j,k)/sqrt( Mijk(c,2,0,0)^(i+j+k) ):
end:

#Pos(c): the integral of the distirbution normalized exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c) over x,y,z>=0
Pos:=proc(c) local lu,a,b,x,y,z:
lu:=exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c):

a:=int(int(int(lu,x=-infinity..infinity),y=-infinity..infinity),z=-infinity..infinity):

b:=evalf(Int(Int(Int(lu,x=0..infinity),y=0..infinity),z=0..infinity)):

b/a:
end:


#EZ(x,y,z,c):exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c):
EZ:=proc(x,y,z,c):
exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c):
end:


#ProbSucker(a,b,c): the probability that a random a-face die, b-faced die, and c-faced die, with all different labels on the
#a+b+c faces is  a sucker-bet's triple. Try:
#ProbSucker(3,3,3);
ProbSucker:=proc(a,b,c) local q,gu,i1,i2,i3,gu1,gu2,gu3,mu:
 gu:=GFq(a,b,c,q):

mu:=0:
for i1  from 1 to degree(gu,q[1]) do
 gu1:=coeff(gu,q[1],i1):
 for i2  from 1 to degree(gu1,q[2]) do
   gu2:=coeff(gu1,q[2],i2):
   
   for i3 from 1 to degree(gu2,q[3]) do
    gu3:=coeff(gu2,q[3],i3):
    mu:=mu+gu3:
    od:
 od:
od:
mu:

end:


#MijkS(c,i,j,k): the (i,j,k) moment of exp(-x^2/2-y^2/-z^2/2 -(x*y+x*z+y*z)*c) normalized. for SYMBOLIC c.
#Try:
#MijkS(c,3,3,3);
MijkS:=proc(c,i,j,k) local lu,a,b,x,y,z:
assume(c>0 and c<1):
lu:=exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c):
a:=int(int(int(lu,x=-infinity..infinity),y=-infinity..infinity),z=-infinity..infinity):

b:=int(int(int(x^i*y^j*z^k*lu,x=-infinity..infinity),y=-infinity..infinity),z=-infinity..infinity):
simplify(b/a):
end:

#SijkS(c,i,j,k): the (i,j,k) standardized moment of exp(-x^2/2-y^2/-z^2/2 -(x*y+x*z+y*z)*c) normalized for symbolic c
SijkS:=proc(c,i,j,k) local a,mu,gu:
option remember:
if i=0 and j=0 and k=0 then
 RETURN(1):
fi:

if i+j+k mod 2=1 then
  RETURN(0):
fi:

mu:=sort([i,j,k]):

if mu<>[i,j,k] then
SijkS(c,op(mu)):
else
gu:=GuessRat([seq([1/a, Sijk(1/a,op(mu))],a=2..2*(i+j+k)+5)],c):
 if gu<>FAIL then
   RETURN(gu):
 else
 RETURN(GuessRatg([seq([1/a, Sijk(1/a,op(mu))],a=2..2*(i+j+k)+3)],c)):
fi:
fi:
end:






#TavlaMom(N): a list-of-lists-of-lists giving SijkS(c,i,j,k) at c=1
#for 0<=i,j,k<=N. Try:
#TavlaMom(2);
TavlaMom:=proc(N) local i,j,k,c:

subs(c=1,[seq([seq([seq(SijkS(c,i,j,k),k=0..N)],j=0..N)],i=0..N)]):

end:

#TavlaMomc(N,c): a list-of-lists-of-lists giving SijkS(c,i,j,k) 
#for 0<=i,j,k<=N. Try:
#TavlaMomc(2,c);
TavlaMomc:=proc(N,c) local i,j,k:

[seq([seq([seq(SijkS(c,i,j,k),k=0..N)],j=0..N)],i=0..N)]:

end:

#TavlaLimSM(N): a list-of-lists-of-lists giving  LimSM
#for 0<=i,j,k<=N. Try:
#TavlaLimSM(2);
TavlaLimSM:=proc(N) local i,j,k:
[seq([seq([seq(LimSM(i,j,k),k=0..N)],j=0..N)],i=0..N)]:
end:

#TavlaLimSMpc(N): a pre-computed version of TavlaLimSM(N).Try:
#TavlaLimSMpc(2);
TavlaLimSMpc:=proc(N) local i,j,k:
[seq([seq([seq(LimSMpc(i,j,k),k=0..N)],j=0..N)],i=0..N)]:
end:


#MomFor(N,k) inputs a pos. integer N and a variable name k, outputs a 
#set of successful (a,b) and a table T, such that T[[a,b]] is
#an explicit formula, in k, for the (a,b,2*k) mixed moment when a+b is even and for the (a,b,2*k-1) mixed moment if a+b is odd.
#Try:
#MomFor(16,k);
MomFor:=proc(N,k) local i,su,gu,a,b,T,lu,mu,P:
lu:=(2*k)!/k!/2^k:

gu:=TavlaMomPC(16):

su:={}:

for a from 0 to N do
 for b from a to N do
 if a+b mod 2=0 then
  mu:=[seq([i,gu[a+1][b+1][2*i+1]/eval(subs(k=i,lu))],i=0..8)]:
 else
  mu:=[seq([i,gu[a+1][b+1][1+2*i-1]/eval(subs(k=i,lu))],i=1..8)]:
fi:


P:=GuessPol1g(mu,k,trunc(N/2)-1):


if P<>FAIL then
 su:=su union {[a,b]}:
 T[[a,b]]:=P:
fi:
od:
od:
su,op(T):
end:




#MomForV(N,k): Verbose form of MomFor(N,k) (q.v). N is a pos. integer (up to 16, k is a variable name). Try:
#MomForV(15,k);
MomForV:=proc(N,k) local gu,a,b,n,c,x,y,z,P,t0:
t0:=time():
if N>16 then
 print(`N must be <=16`):
 RETURN(FAIL):
fi:
gu:=MomFor(15,k):


print(`Explicit Expressions for the Mixed-Moments of The Ekhad-Zeilberger Tri-variate distribution, M(a,b,k) for`):
print(`a+b <=`,N-3, `and arbitrary k `):
print(``):
print(` By Shalosh B. Ekhad `):
print(``):

print(`In this article we will give explicit expressions for the mixed moments, M(a,b,k) for a+b <=`, n-3):
print(`and arbitrary `, k, `for the so-called Ekhad-Zeilberge distribution that is the limit as c->1 from below`):
print(`of the tri-variate distribution whose prob. density function is `):
print(``):
print(EZ(x,y,z,c)):
print(``):

print(`Note that M(a,b,c)=0 if a+b+c is odd`):
print(``):

for a from 0 to N-3 do
 for b from a to N-3 do
   if member([a,b],gu[1]) then
    P:=gu[2][[a,b]]:
    if  a+b mod 2=0 then
     print(``):
     print(M(a,b,2*k)=(2*k)!/2^k/k!*P):
     print(``):
    else
    print(``):
      print(M(a,b,2*k-1)=(2*k)!/2^k/k!*P):
     print(``):
    fi:
  fi:
 od:
od:

print(``):
print(`This ends this paper that took`, time()-t0, `seconds to generate. `):

end:

#MomForV1(N,k): Like MomForV(N,k) but with lprint instead of print. Try:
#MomForV1(15,k);
MomForV1:=proc(N,k) local gu,a,b,n,c,x,y,z,P,t0:
t0:=time():
if N>16 then
 print(`N must be <=16`):
 RETURN(FAIL):
fi:
gu:=MomFor(15,k):

print(`Explicit Expressions for the Mixed-Moments of The Ekhad-Zeilberger Tri-variate distribution, M(a,b,k) for`):
print(`a+b <=`,N-3, `and arbitrary k `):
print(``):
print(` By Shalosh B. Ekhad `):
print(``):

print(`In this article we will give explicit expressions for the mixed moments, M(a,b,k) for a+b <=`, n-3):
print(`and arbitrary `, k, `for the so-called Ekhad-Zeilberge distribution that is the limit as c->1 from below`):
print(`of the tri-variate distribution whose prob. density function is `):
print(``):
print(EZ(x,y,z,c)):
print(``):

print(`Note that M(a,b,c)=0 if a+b+c is odd`):
print(``):
for a from 0 to N-3 do
 for b from a to N-3 do
   if member([a,b],gu[1]) then
    P:=gu[2][[a,b]]:
    if  a+b mod 2=0 then
 print(``):
     lprint(M(a,b,2*k)=(2*k)!/2^k/k!*P):
  print(``):
    else
 print(``):
      lprint(M(a,b,2*k-1)=(2*k)!/2^k/k!*P):
 print(``):
    fi:
  fi:
 od:
od:

print(``):
print(`This ends this paper that took`, time()-t0, `seconds to generate. `):

end:


#MomForVlatex(N,k): Like MomForV(N,k) but with latex instead of print. Try:
#MomForVlatex(15,k);
MomForVlatex:=proc(N,k) local gu,a,b,n,c,x,y,z,P,t0:
t0:=time():
if N>16 then
 print(`N must be <=16`):
 RETURN(FAIL):
fi:
gu:=MomFor(15,k):

print(`Explicit Expressions for the Mixed-Moments of The Ekhad-Zeilberger Tri-variate distribution, M(a,b,k) for`):
print(`a+b <=`,N-3, `and arbitrary k `):
print(``):
print(` By Shalosh B. Ekhad `):
print(``):

print(`In this article we will give explicit expressions for the mixed moments, M(a,b,k) for a+b <=`, n-3):
print(`and arbitrary `, k, `for the so-called Ekhad-Zeilberge distribution that is the limit as c->1 from below`):
print(`of the tri-variate distribution whose prob. density function is `):
print(``):
print(EZ(x,y,z,c)):
print(``):
print(`Note that M(a,b,c)=0 if a+b+c is odd`):
print(``):

for a from 0 to N-3 do
 for b from a to N-3 do
   if member([a,b],gu[1]) then
    P:=gu[2][[a,b]]:
    if  a+b mod 2=0 then

     print(``):
     lprint(latex(M(a,b,2*k)=(2*k)!/2^k/k!*P)):
   print(``):
    else
  print(``):
      lprint(latex(M(a,b,2*k-1)=(2*k)!/2^k/k!*P)):
  print(``):
    fi:
  fi:
 od:
od:

print(``):
print(`This ends this paper that took`, time()-t0, `seconds to generate. `):

end:



#TavlaMfor(N,n) a table of the mixed moments for the 3-Deck Sucker-Bets statistics with three decks and n cards in each
#(with all distinct 3*n cards). Try:
#TavlaMfor(2,n);
TavlaMfor:=proc(N,n) local i,j,k:
[seq([seq([seq(Mfor(n,i,j,k),k=0..N)],j=0..N)],i=0..N)];
end:


#CMFdiag(a,b,c): the diagonal values of the tri-variate cumulative distribution function for the Sucker Bets statistics
#with three decks and deck-sizes a,b,c
# from [0,0,0], incremented by [1,1,1]. Try:
#CMFdiag(5,5,5);
CMFdiag:=proc(a,b,c) local q,gu,i,j,k,A,L1,L2,L3,L:
gu:=GFq(a,b,c,q):
L1:=ldegree(gu,q[1]):
L2:=ldegree(gu,q[2]):
L3:=ldegree(gu,q[3]):

L:=degree(gu,{q[1],q[2],q[3]}):

[seq(add(add(add(coeff(coeff(coeff(gu,q[1],i),q[2],j),q[3],k), i=L1..A),j=L2..A),k=L3..A),A=0..L)]:

end:

#VerifySBC(N): verifies (rigorously) that all the limits of the mixed moments, M(i,j,k)
#(for 0<=i,j,k<=N), of Sucker-Bets distribution with 3 decks and
#n cards in each deck (with all cards differently numbered) as n goes to infinity coincide with the corresponding
#mixed moments of the Ekhad-Zeilberger tri-variate distribution that is the limit as c->1 (from the left) of the 
#distribution whose joint p.df. is exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c):
#For N<=5 it is very fast, since it uses pre-computed values. For N>5 it is very slow. Try:
#VerifySBC(5);
VerifySBC:=proc(N):
evalb(TavlaLimSMpc(N)=TavlaMomPC(N)):
end:







#OpeE(a1,a2,a3,A3,c): The recurrence operator annihilating the scaled (a1,a2,2*a3) moment of the distribution
#EZ(x,y,z,c)=exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c): Try:
#OpeE(a1,a2,a3,A3,c):
OpeE:=proc(a1,a2,a3,A3,c)
option remember:
-4*(-1+c)*(1+2*c)^3*(2*a3+3)*(2*a3+1)*(a3+2)*(a3+1)/(c+1)^5-4*(1+2*c)^2*(a3+2)*
(2*a3+3)*(-2*a3*c^3+a1*c^2+a2*c^2-4*a3*c^2-4*c^3+4*a3*c-6*c^2+4*a3+7*c+7)/(c+1)
^5*A3+(1+2*c)*(a1^2*c^4-2*a1*a2*c^4+a2^2*c^4-4*a3^2*c^4+8*a1*a3*c^3+8*a2*a3*c^3
-24*a3^2*c^3-20*a3*c^4+8*a1*a3*c^2+18*a1*c^3+8*a2*a3*c^2+18*a2*c^3-104*a3*c^3-\
25*c^4+18*a1*c^2+18*a2*c^2+48*a3^2*c+8*a3*c^2-114*c^3+24*a3^2+216*a3*c+18*c^2+
108*a3+246*c+123)/(c+1)^4*A3^2-2*(a1*c^3+a2*c^3-2*a3*c^3+a1*c^2+a2*c^2-5*c^3+8*
a3*c+c^2+4*a3+22*c+11)/(c+1)^2*A3^3+A3^4:
end:


#OpeO(a1,a2,a3,A3,c): The recurrence operator annihilating the scaled (a1,a2,2*a3+1) moment of the distribution
#EZ(x,y,z,c)=exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c): Try:
#OpeO(a1,a2,a3,A3,c):
OpeO:=proc(a1,a2,a3,A3,c)
option remember:
-4*(-1+c)*(1+2*c)^3*(2*a3+5)*(2*a3+3)*(a3+2)*(a3+1)/(c+1)^5-4*(1+2*c)^2*(2*a3+5
)*(a3+2)*(-2*a3*c^3+a1*c^2+a2*c^2-4*a3*c^2-5*c^3+4*a3*c-8*c^2+4*a3+9*c+9)/(c+1)
^5*A3+(1+2*c)*(a1^2*c^4-2*a1*a2*c^4+a2^2*c^4-4*a3^2*c^4+8*a1*a3*c^3+8*a2*a3*c^3
-24*a3^2*c^3-24*a3*c^4+8*a1*a3*c^2+22*a1*c^3+8*a2*a3*c^2+22*a2*c^3-128*a3*c^3-\
36*c^4+22*a1*c^2+22*a2*c^2+48*a3^2*c+8*a3*c^2-172*c^3+24*a3^2+264*a3*c+22*c^2+
132*a3+366*c+183)/(c+1)^4*A3^2-2*(a1*c^3+a2*c^3-2*a3*c^3+a1*c^2+a2*c^2-6*c^3+8*
a3*c+c^2+4*a3+26*c+13)/(c+1)^2*A3^3+A3^4:
end:

#OpeE1(a1,a2,a3,A3,c): the recurrence OpeE(a1,a2,a3,A3,c) in descending powers and in explicit form. Try:
#
OpeE1:=proc(a1,a2,a3,A3,c) local ope:
option remember:
ope:=OpeE(a1,a2,a3,A3,c):
ope:=expand(1-subs(a3=a3-4,ope)/A3^4):
add(factor(coeff(ope,A3,-i))*A3^(-i),i=1..4):
end:

#OpeO1(a1,a2,a3,A3,c): the recurrence OpeO(a1,a2,a3,A3,c) in descending powers and in explicit form. Try:
#
OpeO1:=proc(a1,a2,a3,A3,c) local ope:
option remember:
ope:=OpeO(a1,a2,a3,A3,c):
ope:=expand(1-subs(a3=a3-4,ope)/A3^4):
add(factor(coeff(ope,A3,-i))*A3^(-i),i=1..4):
end:



#CheckOpeE1(): checks the validity of OpeE1. Try:
#CheckOpeE1();
CheckOpeE1:=proc() local a1,a2,a3,c,A3,ope,i,b1,b2,b3,gu:
gu:=TavlaMomcPC(16,c):
ope:=OpeE1(a1,a2,a3,A3,c):
evalb(
{seq(seq(seq( normal(add(subs({a1=b1,a2=b2,a3=b3},coeff(ope,A3,-i))*gu[b1+1][b2+1][2*b3+1-2*i],i=1..4) -gu[b1+1][b2+1][2*b3+1]
 ),b1=0..16),b2=0..16),b3=4..8)}
={0}):
end:

#CheckOpeE(): checks the validity of OpeE. Try:
#CheckOpeE();
CheckOpeE:=proc() local a1,a2,a3,c,A3,ope,i,b1,b2,b3,gu:
gu:=TavlaMomcPC(16,c):
ope:=OpeE(a1,a2,a3,A3,c):
evalb(
{seq(seq(seq(normal(add(subs({a1=b1,a2=b2,a3=b3},coeff(ope,A3,i))*gu[b1+1][b2+1][2*b3+2*i+1],i=0..4)),b1=0..16),b2=0..16),b3=0..4)}
={0}):
end:

#CheckOpeO(): checks the validity of OpeE. Try:
#CheckOpeO();
CheckOpeO:=proc() local a1,a2,a3,c,A3,ope,i,b1,b2,b3,gu:
gu:=TavlaMomcPC(16,c):
ope:=OpeO(a1,a2,a3,A3,c):
evalb(
{seq(seq(seq(normal(add(subs({a1=b1,a2=b2,a3=b3},coeff(ope,A3,i))*gu[b1+1][b2+1][2*b3+2*i+2],i=0..4)),b1=0..16),b2=0..16),b3=0..3)}
={0}):
end:


#CheckOpeO1(): checks the validity of OpeO1. Try:
#CheckOpeO1();
CheckOpeO1:=proc() local a1,a2,a3,c,A3,ope,i,b1,b2,b3,gu:
gu:=TavlaMomcPC(16,c):
ope:=OpeO1(a1,a2,a3,A3,c):
evalb(
{seq(seq(seq( normal(add(subs({a1=b1,a2=b2,a3=b3},coeff(ope,A3,-i))*gu[b1+1][b2+1][2*b3+2-2*i],i=1..4) -gu[b1+1][b2+1][2*b3+2]
 ),b1=0..16),b2=0..16),b3=4..7)}
={0}):
end:

#MomEZ(b1,b2,b3,c): The (b1,b2,b3) scaled mixed moment of EZ(x,y,z,c) done via the recurrence obtained from the
#MultiAlmkvistZeilberger package of Apagodu and Zeilbegrger . Try:
#MomEZ(20,20,20,c);
MomEZ:=proc(b1,b2,b3,c) local gu,ope,a1,a2,a3,A3,mu,i:
option remember:
gu:=TavlaMomcPC(8,c):

if b1<=8 and b2<=8 and b3<=8 then
 RETURN(gu[b1+1][b2+1][b3+1]):
fi:

mu:=sort([b1,b2,b3]):

if mu<>[b1,b2,b3] then
 RETURN(MomEZ(op(mu),c)):
fi:

if b1+b2 mod 2=0 then

if b3 mod 2=1 then
  RETURN(0):
else
 ope:=OpeE1(a1,a2,a3,A3,c):
RETURN(normal(add(subs({a1=b1,a2=b2,a3=b3/2},coeff(ope,A3,-i))*MomEZ(b1,b2,b3-2*i,c),i=1..4))):
fi:

else

if b3 mod 2=0 then
  RETURN(0):
else

 ope:=OpeO1(a1,a2,a3,A3,c):

RETURN(normal(add(subs({a1=b1,a2=b2,a3=(b3-1)/2},coeff(ope,A3,-i))*MomEZ(b1,b2,b3-2*i,c),i=1..4))):
fi:

fi:


end:


#CheckMomEZ(N): checks procedure MomEZ up to N (N<=16). Try:
#CheckMomEZ(10);
CheckMomEZ:=proc(N) local b1,b2,b3,c,gu:
gu:=TavlaMomcPC(N,c):  
evalb({seq(seq(seq(normal(MomEZ(b1,b2,b3,c)-gu[b1+1][b2+1][b3+1]),b1=0..N),b2=0..N),b3=0..N)}={0}):
end:


#OpeD(n,N,c): The recurrence operator annihilating the scaled (2n,2n,2n) mixed moment  of the distribution,
#EZ(x,y,z,c)=exp(-x^2/2-y^2/2-z^2/2 -(x*y+x*z+y*z)*c)/sqrt(2*Pi)^3/(c+1)*(1-c^2)*sqrt(1+2*c):
#where N is the shift operator in n. Try:
#OpeD(n,N,c):
OpeD:=proc(n,N,c) :
-4*(2*n+3)*(n+1)^2*(2*n+1)^3*(1+2*c)^3*(-1+c)^3*(2*c*n+3*c+4*n+7)/(c+1)^7/(2*c*
n+c+4*n+3)+(2*n+3)*(8*c^5*n^3+28*c^5*n^2-64*c^4*n^3+30*c^5*n-220*c^4*n^2-256*c^
3*n^3+9*c^5-232*c^4*n-936*c^3*n^2-224*c^2*n^3-69*c^4-1068*c^3*n-832*c^2*n^2-80*
c*n^3-368*c^3-972*c^2*n-296*c*n^2-32*n^3-348*c^2-350*c*n-120*n^2-129*c-144*n-55
)/(c+1)^4/(2*c*n+c+4*n+3)*N+N^2:
end:

#CheckOpeD(): checks the validity of OpeE. Try:
#CheckOpeD();
CheckOpeD:=proc() local n1,n,N,c,gu,ope,i:
gu:=TavlaMomcPC(16,c):
ope:=OpeD(n,N,c):
evalb({seq(normal(add(subs({n=n1},coeff(ope,N,i))*gu[2*n1+1+2*i][2*n1+1+2*i][2*n1+1+2*i],i=0..2)),n1=0..6)}={0}):
end:

#OpeD1(n,N,c): the recurrence operator OpeD(n,N,c) in descending powers and in explicit form. Try:
#OpeD1(n,N,c):
OpeD1:=proc(n,N,c) local ope:
option remember:
ope:=OpeD(n,N,c):
ope:=expand(1-subs(n=n-2,ope)/N^2):
add(factor(coeff(ope,N,-i))*N^(-i),i=1..2):
end:


#CheckOpeD1(): checks the validity of OpeE1. Try:
#CheckOpeD1();
CheckOpeD1:=proc() local n,N,c,n1,gu,ope,i:
gu:=TavlaMomcPC(16,c):
ope:=OpeD1(n,N,c):
evalb(
{seq( normal(add(subs({n=n1},coeff(ope,N,-i))*gu[2*n1+1-2*i][2*n1+1-2*i][2*n1+1-2*i],i=1..2) -gu[2*n1+1][2*n1+1][2*n1+1]
 ),n1=2..8)}
={0}):
end:


#MomEZd(n1,c): The (2n1,2n1,2n1) scaled mixed moment of EZ(x,y,z,c) done via the diagonal recurrence obtained from the
#MultiAlmkvistZeilberger package of Apagodu and Zeilbegrger . Try:
#MomEZd(10,c);
MomEZd:=proc(n1,c) local gu,ope,n,N:
option remember:
gu:=TavlaMomcPC(4,c):

if n1<=1 then
 RETURN(gu[2*n1+1][2*n1+1][2*n1+1]):
fi:
ope:=OpeD1(n,N,c):
normal(subs(n=n1,coeff(ope,N,-1))*MomEZd(n1-1,c)+subs(n=n1,coeff(ope,N,-2))*MomEZd(n1-2,c)):
end:


#MomEZd1(n1) : the explicit formula (3*n1)!*(2*n1)!/8^n1/n1!^2 for the scaled mixed (2n1,2n1,2n1) Ekhad-Zeilbeger moment.
#Same as MomEZd(n1,1) but done via the explicit formula. Try:
#MomEZd1(10);
MomEZd1:=proc(n1) 
(3*n1)!*(2*n1)!/8^n1/n1!^2:
end:

#CheckMomEZd(N): checks procedure MomEZd up to N . Try:
#CheckMomEZd(10);
CheckMomEZd:=proc(N) local n1,c:
evalb({seq(MomEZd(n1,c)-MomEZ(2*n1,2*n1,2*n1,c),n1=0..N)}={0}):
end:

#CheckMomEZd1(N): checks procedure MomEZd1 up to N . Try:
#CheckMomEZd1(10);
CheckMomEZd1:=proc(N) local n1:
evalb({seq(MomEZd1(n1)-MomEZd(n1,1),n1=0..N)}={0}):
end:


#SBpaper(N): An article about the Sucker-Bets Statistics, giving all the mixed-moments up to the (N,N,N)
#and verifying that the limits of scaled limits are those of the corresponding one for EZ(x,y,z,c) as c goes to 1. Try:`)
#SBpaper(5);
SBpaper:=proc(N) local c,n,mu,lu,i,j,k,x,y,z,M,ope1e,ope1o,A3,a1,a2,a3:

if not evalb(TavlaLimSMpc(N)=TavlaMomPC(N)) then
 print(`The conjecture is false`):
 RETURN(false):
fi:

mu:=TavlaMforPC(N,n):
lu:=TavlaLimSMpc(N):

print(`Explicit [rigorously-proved] (polynomial) Formuals for all the Mixed Moments up to the (`, N,N,N, `) `):
print(`of the Sucker-Bets Distribution with 3 Decks,  each with n cards, where all the 3n cards have distinct numbers`):
print(`and a proof that the scaled mixed moments converge, as n goes to infinity, to the corresponding mixed moments of the`):
print(`Ekhad-Zeilberger trivariate continuous distribution, i.e. the limit, as c goes to 1 from below of the distribution`):
print(`whose probability density function is`):
print(``):
print(EZ(x,y,z,c)):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

print(`Consider the set of all`, (3*n)!/n!^3,  `words in the alphabet {1,2,3} with n 1's, n 2's and n 3's `):
print(`note that from each such word we can form a set of three card-decks, each with n cards,with denominations from 1 to 3n`):
print(`by forming the first deck with cards corresponding to the locations of 1`):
print(`the second  deck with cards corresponding to the locations of 2`):
print(`the third  deck with cards corresponding to the locations of 3`):

print(`and define three statistics, S12, S23, S31, where `):
print(``):
print(` S12(w)= (Number of occurrences of 1 before 2) Minus (Number of occurrences of 2 before 1)`):
print(``):
print(` S23(w)= (Number of occurrences of 2 before 3) Minus (Number of occurrences of 3 before 2)`):
print(``):
print(` S31(w)= (Number of occurrences of 3 before 1) Minus (Number of occurrences of 1 before 3)`):
print(``):
print(`Note that "before" does not (necessarily) means "right before"`):

print(`Note that the triples of decks of cards corresponding to words for which all of S12(w),S23(w), S31(w) are strictly positive are sucker's bets`):

print(`The following are explicit expressions for all the mixed moments M(i,j,k), for 0<=i<=j<=k<=5, and i+j+k even`):
print(`Note that it is always 0 when i+j+k is odd, and by symmetry you can get all of them. `):


print(`The variance of each of S12, S23, S31 is`):
print(``):
print(mu[3][1][1]):
print(``):

print(`The fourth-moment of each of S12, S23, S31 is`):
print(``):
print(mu[5][1][1]):
print(``):
print(`Hence the kurtosis is`):
print(``):
print(mu[5][1][1]/mu[3][1][1]^2):
print(``):
print(`whose limit, as n goes to infinity is`):
print(``):
print(lu[5][1][1]):
print(``):
print(`as it should!, since they are each asymptotically normal.`):
print(`However they are far from independent!`):
print(``):
print(`The covariance between any two of them is`):
print(``):
print(mu[2][2][1]):
print(``):
print(`Hence the correlation coefficient between any two of these random variables is`):
print(``):
print(mu[2][2][1]/mu[3][1][1]):
print(``):
print(`whose limit is`):
print(``):
print(lu[2][2][1]):
print(``):
print(`Let M(i,j,k) be the (i,j,k) mixed moment for the statistics S12(w), S23(w), and S31(w) `):
print(``):
for i from 0 to N do
 for j from i to N do
   for k from j to N do
    if i+j+k mod 2=0 then
     print(M(i,j,k)=mu[i+1][j+1][k+1] ):
    fi:
   od:
 od:
od:

print(`It follows that the scaled mixed moments, M(i,j,k)/M(2,0,0)^((i+j+k)/2)), as n goes to infinity, let's call these`):
print(`limits S(i,j,k), are as follows.`):

for i from 0 to N do
 for j from i to N do
   for k from j to N do
    if i+j+k mod 2=0 then
     print(S(i,j,k)=lu[i+1][j+1][k+1] ):
    fi:
   od:
 od:
od:

ope1e:=OpeE1(a1,a2,a3,A3,c):
ope1o:=OpeO1(a1,a2,a3,A3,c):

print(``):
print(`Consider the trivariate distribution whose probability density function in x,y,z, is`):
print(``):
print(EZ(x,y,z,c)):
print(``):
print(`where c is between 0 and 1. `):
print(``):
print(`We claim that the limit as c goes to 1, from below of the above tri-variate continuous distribution`):
print(`is the  asymptotic limit of the (scaled) Sucker-Bets tri-variate statistics as n goes to infinity`):
print(``):
print(`Let Mc(a1,a2,a3) be its scaled (i,j,k)-mixed moment (i.e. the (a1,a2,a3) mixed moment, divided by the standard deviation to the power`):
print(`a1+a2+a3 `):
print(``):
print(`Using the amazing Apagodu-Zeilberger multi-variate Alkvist-Zeilberger algorithm the following theorem was discovered`):
print(`and proved (but we omit the proof)`):
print(``):
print(`Theorem: If a1+a2 is even we have the following recurrence `):
print(``):
print(Mc(a1,a2,2*a3)=add(coeff(ope1e,A3,-i)*Mc(a1,a2,2*a3-2*i),i=1..4)):
print(``):

print(`If a1+a2 is odd we have the following recurrence `):
print(``):
print(Mc(a1,a2,2*a3+1)=add(coeff(ope1o,A3,-i)*Mc(a1,a2,2*a3-2*i+1),i=1..4)):
print(``):

print(`Note that this enables a very fast computation of these mixed moments`):
print(``):
print(`Taking the limits as c goes to  1 (from below),let's call  S(a1,a2,a3)=Mc(a1,a2,a3,1), we get the following corollary`):
print(``):

ope1e:=subs(c=1,ope1e):
ope1o:=subs(c=1,ope1o):
ope1e:=add(factor(coeff(ope1e,A3,-i))*A3^(-i),i=1..-ldegree(ope1e,A3)):
ope1o:=add(factor(coeff(ope1e,A3,-i))*A3^(-i),i=1..-ldegree(ope1o,A3)):


print(`Corollary: If a1+a2 is even we have the following recurrence `):
print(``):
print(S(a1,a2,2*a3)=add(coeff(ope1e,A3,-i)*S(a1,a2,2*a3-2*i),i=1..4)):
print(``):

print(`If a1+a2 is odd we have the following recurrence `):
print(``):
print(S(a1,a2,2*a3+1)=add(coeff(ope1o,A3,-i)*S(a1,a2,2*a3-2*i+1),i=1..4)):
print(``):
print(`This enables very fast computation of as many mixed moments as we wish.`):
print(`In particular it confirms our conjecture that up to all mixed moments (i,j,k) with, i,j,k <=`, N):
print(`the limiting distribution of the 3-Deck Sucker Bets statistics (each with n cards) tends to this one`):
print(``):
print(`The mixed moments, even for c=1, do not have closed form, but the diagonal, S(2n,2n,2n) surprisingly do!`):
print(``):
print(`The following theorem also follows from the Apagodu-Zeilberger multi-variate Alkvist-Zeilberger algorithm`):
print(``):
print(`Theorem: for each non-negative integer, we have`):
print(``):
print(S(2*n,2*n,2*n)=MomEZd1(n)):
print(``):
print(`This ends this article. `):

end:


#SBpaperLP(N): Like SBpaper(N) (q.v.) but outputting it in Maple input format so that it can be used. Try:
#SBpaperLP(5);
SBpaperLP:=proc(N) local c,n,mu,lu,i,j,k,x,y,z,M,ope1e,ope1o,A3,a1,a2,a3:

if not evalb(TavlaLimSMpc(N)=TavlaMomPC(N)) then
 print(`The conjecture is false`):
 RETURN(false):
fi:

mu:=TavlaMforPC(N,n):
lu:=TavlaLimSMpc(N):

print(`Explicit [rigorously-proved] (polynomial) Formuals for all the Mixed Moments up to the (`, N,N,N, `) `):
print(`of the Sucker-Bets Distribution with 3 Decks,  each with n cards, where all the 3n cards have distinct numbers`):
print(`and a proof that the scaled mixed moments converge, as n goes to infinity, to the corresponding mixed moments of the`):
print(`Ekhad-Zeilberger trivariate continuous distribution, i.e. the limit, as c goes to 1 from below of the distribution`):
print(`whose probability density function is`):
print(``):
lprint(EZ(x,y,z,c)):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

print(`Consider the set of all`, (3*n)!/n!^3,  `words in the alphabet {1,2,3} with n 1's, n 2's and n 3's `):
print(`note that from each such word we can form a set of three card-decks, each with n cards,with denominations from 1 to 3n`):
print(`by forming the first deck with cards corresponding to the locations of 1`):
print(`the second  deck with cards corresponding to the locations of 2`):
print(`the third  deck with cards corresponding to the locations of 3`):

print(`and define three statistics, S12, S23, S31, where `):
print(``):
print(` S12(w)= (Number of occurrences of 1 before 2) Minus (Number of occurrences of 2 before 1)`):
print(``):
print(` S23(w)= (Number of occurrences of 2 before 3) Minus (Number of occurrences of 3 before 2)`):
print(``):
print(` S31(w)= (Number of occurrences of 3 before 1) Minus (Number of occurrences of 1 before 3)`):
print(``):
print(`Note that "before" does not (necessarily) means "right before"`):

print(`Note that the triples of decks of cards corresponding to words for which all of S12(w),S23(w), S31(w) are strictly positive are sucker's bets`):

print(`The following are explicit expressions for all the mixed moments M(i,j,k), for 0<=i<=j<=k<=5, and i+j+k even`):
print(`Note that it is always 0 when i+j+k is odd, and by symmetry you can get all of them. `):


print(`The variance of each of S12, S23, S31 is`):
print(``):
lprint(mu[3][1][1]):
print(``):

print(`The fourth-moment of each of S12, S23, S31 is`):
print(``):
lprint(mu[5][1][1]):
print(``):
print(`Hence the kurtosis is`):
print(``):
lprint(mu[5][1][1]/mu[3][1][1]^2):
print(``):
print(`whose limit, as n goes to infinity is`):
print(``):
lprint(lu[5][1][1]):
print(``):
print(`as it should!, since they are each asymptotically normal.`):
print(`However they are far from independent!`):
print(``):
print(`The covariance between any two of them is`):
print(``):
lprint(mu[2][2][1]):
print(``):
print(`Hence the correlation coefficient between any two of these random variables is`):
print(``):
lprint(mu[2][2][1]/mu[3][1][1]):
print(``):
print(`whose limit is`):
print(``):
lprint(lu[2][2][1]):
print(``):
print(`Let M(i,j,k) be the (i,j,k) mixed moment for the statistics S12(w), S23(w), and S31(w) `):
print(``):
for i from 0 to N do
 for j from i to N do
   for k from j to N do
    if i+j+k mod 2=0 then
     lprint(M(i,j,k)=mu[i+1][j+1][k+1] ):
    fi:
   od:
 od:
od:

print(`It follows that the scaled mixed moments, M(i,j,k)/M(2,0,0)^((i+j+k)/2)), as n goes to infinity, let's call these`):
print(`limits S(i,j,k), are as follows.`):

for i from 0 to N do
 for j from i to N do
   for k from j to N do
    if i+j+k mod 2=0 then
     lprint(S(i,j,k)=lu[i+1][j+1][k+1] ):
    fi:
   od:
 od:
od:

ope1e:=OpeE1(a1,a2,a3,A3,c):
ope1o:=OpeO1(a1,a2,a3,A3,c):

print(``):
print(`Consider the trivariate distribution whose probability density function in x,y,z, is`):
print(``):
lprint(EZ(x,y,z,c)):
print(``):
print(`where c is between 0 and 1. `):
print(``):
print(`We claim that the limit as c goes to 1, from below of the above tri-variate continuous distribution`):
print(`is the  asymptotic limit of the (scaled) Sucker-Bets tri-variate statistics as n goes to infinity`):
print(``):
print(`Let Mc(a1,a2,a3) be its scaled (i,j,k)-mixed moment (i.e. the (a1,a2,a3) mixed moment, divided by the standard deviation to the power`):
print(`a1+a2+a3 `):
print(``):
print(`Using the amazing Apagodu-Zeilberger multi-variate Alkvist-Zeilberger algorithm the following theorem was discovered`):
print(`and proved (but we omit the proof)`):
print(``):
print(`Theorem: If a1+a2 is even we have the following recurrence `):
print(``):
lprint(Mc(a1,a2,2*a3)=add(coeff(ope1e,A3,-i)*Mc(a1,a2,2*a3-2*i),i=1..4)):
print(``):

print(`If a1+a2 is odd we have the following recurrence `):
print(``):
lprint(Mc(a1,a2,2*a3+1)=add(coeff(ope1o,A3,-i)*Mc(a1,a2,2*a3-2*i+1),i=1..4)):
print(``):

print(`Note that this enables a very fast computation of these mixed moments`):
print(``):
print(`Taking the limits as c goes to  1 (from below),let's call Mc(a1,a2,a3,1), S(a1,a2,a3), we get the following corollary`):
print(``):

ope1e:=subs(c=1,ope1e):
ope1o:=subs(c=1,ope1o):
ope1e:=add(factor(coeff(ope1e,A3,-i))*A3^(-i),i=1..-ldegree(ope1e,A3)):
ope1o:=add(factor(coeff(ope1e,A3,-i))*A3^(-i),i=1..-ldegree(ope1o,A3)):


print(`Corollary: If a1+a2 is even we have the following recurrence `):
print(``):
print(S(a1,a2,2*a3)=add(coeff(ope1e,A3,-i)*S(a1,a2,2*a3-2*i),i=1..4)):
print(``):

print(`If a1+a2 is odd we have the following recurrence `):
print(``):
lprint(S(a1,a2,2*a3+1)=add(coeff(ope1o,A3,-i)*S(a1,a2,2*a3-2*i+1),i=1..4)):
print(``):
print(`This enables very fast computation of as many mixed moments as we wish.`):
print(`In particular it confirms our conjecture that up to all mixed moments (i,j,k) with, i,j,k <=`, N):
print(`the limiting distribution of the 3-Deck Sucker Bets statistics (each with n cards) tends to this one`):
print(``):
print(`The mixed moments, even for c=1, do not have closed form, but the diagonal, S(2n,2n,2n) surprisingly do!`):
print(``):
print(`The following theorem also follows from the Apagodu-Zeilberger multi-variate Alkvist-Zeilberger algorithm`):
print(``):
print(`Theorem: for each non-negative integer, we have`):
print(``):
lprint(S(2*n,2*n,2*n)=MomEZd1(n)):
print(``):
print(`This ends this article. `):

end:



#SBseq(N): the sequence of length N  whose n-th entry is the number of sucker bets with each of the decks with n cards
#or equivalently, the number of words w, in the alphabet {1,2,3} with n 1's n 2's and n 3's such that
#S12(w)>0, S23(w), and S31(w)>0. Try: 
#SBseq(5);
SBseq:=proc(N) local n:
[seq(ProbSucker(n,n,n)*(3*n)!/n!^3,n=1..N)]:
end: