###################################################################### ## Berele.txt Save this file as Berele.txt to use it, # # stay in the # ## same directory, get into Maple (by typing: maple ) # ## and then type: read `Berele.txt` # ## Then follow the instructions given there # ## # ## Written by Doron Zeilberger, Rutgers University , # ## DoronZeil at gmail dot com # ###################################################################### print(`First Written: June 2022: tested for Maple 2020 `): print(): print(`This is Berele.txt, A Maple package`): print(`accompanying Shalsoh B. Ekhad and Doron Zeilberger's article: `): print(`Fast Computations of Certain Poincare Series Introduced by Allan Berele in the Theory of Polynomial Identity Rings and Invariant Theory `): print(): print(`The most current version is available on WWW at:`): print(` http://sites.math.rutgers.edu/~zeilberg/tokhniot/Berele.txt .`): print(`Please report all bugs to: DoronZeil at gmail dot com .`): print(): print(`------------------------------------`): print(`For general help, and a list of the MAIN functions,`): print(` type "ezra();". For specific help type "ezra(procedure_name);" `): print(`------------------------------------`): print(`For a list of the supporting functions type: ezra1();`): print(` For specific help type "ezra(procedure_name);" `): print(): print(`------------------------------------`): print(`For a list of the STORY functions type: ezraS();`): print(` For specific help type "ezra(procedure_name);" `): print(): print(`------------------------------------`): ezra1:=proc() if args=NULL then print(`The SUPPORTING, or marginal, procedures are`): print(` Bn, BnTaylor, BnTry, Bp, Bp2, Bp3, Bn, Bn2, Bn3, CheckBn3 `): else ezra(args): fi: end: ezraS:=proc() if args=NULL then print(`The STORY procedures are`): print(` Paper2, Paper3 `): else ezra(args): fi: end: ezra:=proc() if args=NULL then print(` Berele.txt: A Maple package for computing Poincare series of interest to Allan Berele `): print(`The MAIN procedures are`): print(` Cnk, C2k, C3k`): elif nargs=1 and args[1]=Bn then print(`Bn(n,k,t):The constant term of mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^(-k),j=i+1..n),i=1..n) done directly. Very slow for n>=3. Try:`): print(`Bn(2,4,t);`): elif nargs=1 and args[1]=BnTaylor then print(`BnTaylor(n,k,t,N):The first N terms in the Taylor expansion of `): print(`constant term of mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^(-k),j=i+1..n),i=1..n). Mostly for checking. Try:`): print(`BnTaylor(3,5,t,10);`): elif nargs=1 and args[1]=Bn2 then print(`Bn2(k,t): A fast way to compute Bn(2,k,t), using the recurrence obtained via the Almkvist-Zeilberger algorithm. try:`): print(`Bn2(30,t);`): elif nargs=1 and args[1]=Bn3 then print(`Bn3(k,t): A fast way to compute Bn(3,k,t), using the recurrence obtained via the Apagodu-Zeilberger multi-Almkvist-Zeilberger algorithm. try:`): print(`Bn3(30,t);`): elif nargs=1 and args[1]=BnTry then print(`BnTry(n,k,t,K):The constant term of mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^(-k),j=i+1..n),i=1..n) done directly. K is a guessing parameter`): elif nargs=1 and args[1]=Bp then print(`Bp(n,k,t):The constant term of mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^k,j=i+1..n),i=1..n) done directly. Warning: VERY slow for n>2 and k>5. Try:`): print(`Bp(2,3,t);`): elif nargs=1 and args[1]=Bp2 then print(`Bp2(k,t): A fast way to compute Bp(2,k,t), using the recurrence obtained via the Almkvist-Zeilberger algorithm. try:`): print(`Bp2(30,t);`): elif nargs=1 and args[1]=Bp3 then print(`Bp3(k,t): A fast way to compute Bp(3,k,t), using the recurrence obtained via the Apagodu-Zeilberger multi-Almkvist-Zeilberger algorithm. try:`): print(`Bp3(30,t);`): elif nargs=1 and args[1]=C3 then print(`C3k(k,t): The Poincare series of Allan Berele's C bar(3,k) in the variable t, done via the recurrence. Much faster. Try:`): print(`C3k(10,t);`): elif nargs=1 and args[1]=CheckBn3 then print(`CheckBn3(k,N): Checks that the first N Taylor coefficients of Bn3(i,t) are the same as those of B(3,i,t) for i=1..k. Try:`): print(`CheckBn3(5,20);`): elif nargs=1 and args[1]=Cnk then print(`Cnk(n,k,t): The Poincare series of Allan Berele's C bar(n,k) in the variable t, done directly. For large n and k it is very slow.Try:`): print(`Cnk(4,1,t);`): elif nargs=1 and args[1]=C2k then print(`C2k(k,t): The Poincare series of Allan Berele's C bar(2,k) in the variable t, done via the recurrence obtained from the Almkist-Zeilberger algorithm. Much faster. Try:`): print(`C2k(10,t);`): elif nargs=1 and args[1]=Cn3 then print(`Cn3(k,t): A fast way to compute Cn(3,k,t), using the recurrence obtained via the Apagodu-Zeilberger multi-Almkvist-Zeilberger algorithm. try:`): print(`Cn3(30,t);`): elif nargs=1 and args[1]=Paper2 then print(`Paper2(K): outputs a paper with the recurrence for the Poincare series of Cbar(2,k) and the first K terms. Try:`): print(`Paper2(30);`): elif nargs=1 and args[1]=Paper3 then print(`Paper3(K): outputs a paper with the recurrence for the Poincare series of Cbar(3,k) and the first K terms. Try:`): print(`Paper3(30);`): else print(`There is no such thing as`, args): fi: end: ###FROM Cfinite.txt #GuessRec1(L,d): inputs a sequence L and tries to guess #a recurrence operator with constant cofficients of order d #satisfying it. It returns the initial d values and the operator # as a list. For example try: #GuessRec1([1,1,1,1,1,1],1); GuessRec1:=proc(L,d) local eq,var,a,i,n: if nops(L)<=2*d+2 then print(`The list must be of size >=`, 2*d+3 ): RETURN(FAIL): fi: var:={seq(a[i],i=1..d)}: eq:={seq(L[n]-add(a[i]*L[n-i],i=1..d),n=d+1..nops(L))}: var:=solve(eq,var): if var=NULL then RETURN(FAIL): else RETURN([[op(1..d,L)],[seq(subs(var,a[i]),i=1..d)]]): fi: end: #GuessRec(L): inputs a sequence L and tries to guess #a recurrence operator with constant cofficients #satisfying it. It returns the initial values and the operator # as a list. For example try: #GuessRec([1,1,1,1,1,1]); GuessRec:=proc(L) local gu,d: for d from 1 to trunc(nops(L)/2)-2 do gu:=GuessRec1(L,d): if gu<>FAIL then RETURN(gu): fi: od: FAIL: end: #SeqFromRec(S,N): Inputs S=[INI,ope] #where INI is the list of initial conditions, a ope a list of #size L, say, and a recurrence operator ope, codes a list of #size L, finds the first N0 terms of the sequence satisfying #the recurrence f(n)=ope[1]*f(n-1)+...ope[L]*f(n-L). #For example, for the first 20 Fibonacci numbers, #try: SeqFromRec([[1,1],[1,1]],20); SeqFromRec:=proc(S,N) local gu,L,n,i,INI,ope: INI:=S[1]:ope:=S[2]: if not type(INI,list) or not type(ope,list) then print(`The first two arguments must be lists `): RETURN(FAIL): fi: L:=nops(INI): if nops(ope)<>L then print(`The first two arguments must be lists of the same size`): RETURN(FAIL): fi: if not type(N,integer) then print(`The third argument must be an integer`, L): RETURN(FAIL): fi: if Nexpand(SeqFromRec(S,L+1)) then print([seq(coeff(f1,t,i),i=0..L)],SeqFromRec(S,L+1)): RETURN(FAIL): else RETURN(f): fi: end: ###End FROM Cfinite.txt #Bp(n,k,t):The constant term of mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^k,j=i+1..n),i=1..n) done directly Bp:=proc(n,k,t) local i,j,gu,z: gu:=mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^k,j=i+1..n),i=1..n): for i from 1 to n do gu:=coeff(gu,z[i],0): od: expand(gu): end: #BnTaylor(n,k,t,N):The first N terms in the Taylor expansion of #constant term of mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^(-k),j=i+1..n),i=1..n) #BnTaylor(n,k,t,N)= the first N terms in the Poincare series in t BnTaylor:=proc(n,k,t,N) local i,j,gu,z: gu:=mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^(-k),j=i+1..n),i=1..n): gu:=taylor(gu,t=0,N+1): gu:=add(expand(coeff(gu,t,i))*t^i,i=0..N): for i from 1 to n do gu:=coeff(gu,z[i],0): od: expand(gu): end: #BnTry(n,k,t,K):The constant term of mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^(-k),j=i+1..n),i=1..n) done directly. K is a guessing parameter BnTry:=proc(n,k,t,K) local gu,lu,i: gu:=BnTaylor(n,k,t,K): lu:=[seq(coeff(gu,t,i),i=0..K)]: lu:=GuessRec(lu): if lu=FAIL then RETURN(FAIL): else RETURN(factor(CtoR(lu,t))): fi: end: #Bn(n,k,t):The constant term of mul(mul( (1-z[i]/z[j])*(1-z[j]/z[i]),j=i+1..n),i=1..n)*mul(mul( ((1-t*z[i]/z[j])*(1-t*z[j]/z[i]))^(-k),j=i+1..n),i=1..n) done directly. Very slow Bn:=proc(n,k,t) local K,gu: for K from 10 by 10 to 1000 do gu:=BnTry(n,k,t,K): if gu<>FAIL then RETURN(gu): fi: od: FAIL: end: #2*(k*t^2-2*t^2+k-t-2)/(t-1)^2/(t+1)^2/(k-1)/K-(-3+k)/(t-1)^2/(t+1)^2/(k-1)/K^2: #[2/(1+t), -2/(t-1)/(1+t)^3] #Bn2(k,t): A fast way to compute Bn(2,k,t) Bn2:=proc(k,t) option remember: if k=0 then RETURN(1): elif k=1 then RETURN(1/(1+t)): elif k=2 then RETURN(-1/(t-1)/(1+t)^3): else normal(2*(k*t^2-2*t^2+k-t-2)/(t-1)^2/(t+1)^2/(k-1)*Bn2(k-1,t)-(-3+k)/(t-1)^2/(t+1)^2/(k-1)*Bn2(k-2,t)): fi: end: Bp3:=proc(k1,t) local INI,ope,k,i,K: option remember: INI:= [6*t^6+12*t^5+18*t^4+18*t^3+18*t^2+12*t+6, 6*t^12+24*t^11+66*t^10+120*t^9+204*t^8+264*t^7+312*t^6+264*t^5+204*t^4+120*t^3+66*t^2+24*t+6, 6*t^18+36*t^17+144*t^16+378*t^15+900*t^14+1656*t^13+2880*t^12+3870*t^11+4950*t^10+5010*t^9+4950*t^8+3870*t^7+2880*t^6+1656*t^5+900*t^4+ 378*t^3+144*t^2+36*t+6, 6*t^24+48*t^23+252*t^22+864*t^21+2640*t^20+6240*t^19+14076*t^18+25008*t^17+43338*t^16+59808*t^15+82836*t^14+91824*t^13+102876*t^12+91824*t^11+82836*t^10+59808*t^9+43338*t^8+25008*t^7+14076*t^6+6240*t^5+2640*t^4+864*t^3+252*t^2+48*t+6]: if k1=0 then RETURN(6): elif k1>=1 and k1<=4 then RETURN(INI[k1]): else ope:= -1/4*(k^10*t^26-18*k^10*t^25-5*k^9*t^26-1364*k^10*t^24+116*k^9*t^25+2*k^8*t^26-\ 17594*k^10*t^23+10252*k^9*t^24-151*k^8*t^25+20*k^7*t^26-101504*k^10*t^22+126140 *k^9*t^23-27918*k^8*t^24-520*k^7*t^25-19*k^6*t^26-362568*k^10*t^21+653424*k^9*t ^22-313184*k^8*t^23+33540*k^7*t^24+1958*k^6*t^25-25*k^5*t^26-1055240*k^10*t^20+ 2127858*k^9*t^21-1312178*k^8*t^22+298279*k^7*t^23-15240*k^6*t^24-2710*k^5*t^25+ 28*k^4*t^26-2430994*k^10*t^19+6078616*k^9*t^20-3427335*k^8*t^21+541232*k^7*t^22 -26000*k^6*t^23-2640*k^5*t^24+1955*k^4*t^25+10*k^3*t^26-4413138*k^10*t^18+ 13363964*k^9*t^19-10162903*k^8*t^20-548831*k^7*t^21+1053608*k^6*t^22-98570*k^5* t^23+4214*k^4*t^24-774*k^3*t^25-12*k^2*t^26-8050366*k^10*t^17+22227602*k^9*t^18 -19718180*k^8*t^19+2577562*k^7*t^20+5281840*k^6*t^21-956308*k^5*t^22+8304*k^4*t ^23-616*k^3*t^24+144*k^2*t^25-10666691*k^10*t^16+43307556*k^9*t^17-23231341*k^8 *t^18-1932297*k^7*t^19+4123714*k^6*t^20-2913782*k^5*t^21-72086*k^4*t^22+32711*k ^3*t^23-300*k^2*t^24-14813492*k^10*t^15+50610383*k^9*t^16-66144637*k^8*t^17-\ 30637870*k^7*t^18+20111334*k^6*t^19+2873140*k^5*t^20-708953*k^4*t^21+238388*k^3 *t^22-10086*k^2*t^23+72*k*t^24-16422544*k^10*t^14+74948296*k^9*t^15-40338195*k^ 8*t^16+21637004*k^7*t^17+62842562*k^6*t^18-6171750*k^5*t^19-5857815*k^4*t^20+ 484087*k^3*t^21-30368*k^2*t^22-17698256*k^10*t^13+78346384*k^9*t^14-94442876*k^ 8*t^15-98484146*k^7*t^16-8090422*k^6*t^17-22614952*k^5*t^18-3537512*k^4*t^19+ 1012290*k^3*t^20+162952*k^2*t^21-11040*k*t^22-16422544*k^10*t^12+86005740*k^9*t ^13-72959467*k^8*t^14-14291230*k^7*t^15+169660921*k^6*t^16+60076694*k^5*t^17-\ 5562727*k^4*t^18+82615*k^3*t^19+682004*k^2*t^20-66468*k*t^21+288*t^22-14813492* k^10*t^11+78346384*k^9*t^12-89804234*k^8*t^13-94830722*k^7*t^14+38601674*k^6*t^ 15-73182431*k^5*t^16-52571679*k^4*t^17+761076*k^3*t^18+492584*k^2*t^19-106056*k *t^20-1152*t^21-10666691*k^10*t^10+74948296*k^9*t^11-72959467*k^8*t^12-73977418 *k^7*t^13+142056502*k^6*t^14+73136912*k^5*t^15+7382049*k^4*t^16+6679874*k^3*t^ 17+1618980*k^2*t^18+243228*k*t^19-20160*t^20-8050366*k^10*t^9+50610383*k^9*t^10 -94442876*k^8*t^11-94830722*k^7*t^12+122775296*k^6*t^13+9266848*k^5*t^14-\ 74393000*k^4*t^15-8147542*k^3*t^16+5383248*k^2*t^17-181488*k*t^18-88272*t^19-\ 4413138*k^10*t^8+43307556*k^9*t^9-40338195*k^8*t^10-14291230*k^7*t^11+142056502 *k^6*t^12+19735628*k^5*t^13-51045343*k^4*t^14+3797410*k^3*t^15+3245116*k^2*t^16 -1145112*k*t^17-51840*t^18-2430994*k^10*t^7+22227602*k^9*t^8-66144637*k^8*t^9-\ 98484146*k^7*t^10+38601674*k^6*t^11+9266848*k^5*t^12-49548950*k^4*t^13-2657862* k^3*t^14+9841342*k^2*t^15+1639320*k*t^16-59184*t^17-1055240*k^10*t^6+13363964*k ^9*t^7-23231341*k^8*t^8+21637004*k^7*t^9+169660921*k^6*t^10+73136912*k^5*t^11-\ 51045343*k^4*t^12-6400198*k^3*t^13+9720932*k^2*t^14-992988*k*t^15-592128*t^16-\ 362568*k^10*t^5+6078616*k^9*t^6-19718180*k^8*t^7-30637870*k^7*t^8-8090422*k^6*t ^9-73182431*k^5*t^10-74393000*k^4*t^11-2657862*k^3*t^12+9568048*k^2*t^13-40776* k*t^14-327600*t^15-101504*k^10*t^4+2127858*k^9*t^5-10162903*k^8*t^6-1932297*k^7 *t^7+62842562*k^6*t^8+60076694*k^5*t^9+7382049*k^4*t^10+3797410*k^3*t^11+ 9720932*k^2*t^12+900984*k*t^13-511200*t^14-17594*k^10*t^3+653424*k^9*t^4-\ 3427335*k^8*t^5+2577562*k^7*t^6+20111334*k^6*t^7-22614952*k^5*t^8-52571679*k^4* t^9-8147542*k^3*t^10+9841342*k^2*t^11-40776*k*t^12-706464*t^13-1364*k^10*t^2+ 126140*k^9*t^3-1312178*k^8*t^4-548831*k^7*t^5+4123714*k^6*t^6-6171750*k^5*t^7-\ 5562727*k^4*t^8+6679874*k^3*t^9+3245116*k^2*t^10-992988*k*t^11-511200*t^12-18*k ^10*t+10252*k^9*t^2-313184*k^8*t^3+541232*k^7*t^4+5281840*k^6*t^5+2873140*k^5*t ^6-3537512*k^4*t^7+761076*k^3*t^8+5383248*k^2*t^9+1639320*k*t^10-327600*t^11+k^ 10+116*k^9*t-27918*k^8*t^2+298279*k^7*t^3+1053608*k^6*t^4-2913782*k^5*t^5-\ 5857815*k^4*t^6+82615*k^3*t^7+1618980*k^2*t^8-1145112*k*t^9-592128*t^10-5*k^9-\ 151*k^8*t+33540*k^7*t^2-26000*k^6*t^3-956308*k^5*t^4-708953*k^4*t^5+1012290*k^3 *t^6+492584*k^2*t^7-181488*k*t^8-59184*t^9+2*k^8-520*k^7*t-15240*k^6*t^2-98570* k^5*t^3-72086*k^4*t^4+484087*k^3*t^5+682004*k^2*t^6+243228*k*t^7-51840*t^8+20*k ^7+1958*k^6*t-2640*k^5*t^2+8304*k^4*t^3+238388*k^3*t^4+162952*k^2*t^5-106056*k* t^6-88272*t^7-19*k^6-2710*k^5*t+4214*k^4*t^2+32711*k^3*t^3-30368*k^2*t^4-66468* k*t^5-20160*t^6-25*k^5+1955*k^4*t-616*k^3*t^2-10086*k^2*t^3-11040*k*t^4-1152*t^ 5+28*k^4-774*k^3*t-300*k^2*t^2+288*t^4+10*k^3+144*k^2*t+72*k*t^2-12*k^2)/(k+1)^ 2/(2*k+1)^2/(k^6*t^18+34*k^6*t^17-7*k^5*t^18+384*k^6*t^16-276*k^5*t^17+15*k^4*t 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22-53206022*k^8*t^13+388588616*k^7*t^14-645526563*k^6*t^15-159371902*k^5*t^16+ 466849732*k^4*t^17+40617409*k^3*t^18-44583521*k^2*t^19+1993332*k*t^20+680616*t^ 21-44406725*k^8*t^12+366522684*k^7*t^13-682323824*k^6*t^14-202259378*k^5*t^15+ 618697565*k^4*t^16+88428502*k^3*t^17-71080406*k^2*t^18+685086*k*t^19+1812240*t^ 20-32799928*k^8*t^11+307460918*k^7*t^12-645526563*k^6*t^13-217834336*k^5*t^14+ 730656176*k^4*t^15+136834784*k^3*t^16-95987629*k^2*t^17-3171756*k*t^18+3530952* t^19-21299821*k^8*t^10+229166362*k^7*t^11-547061782*k^6*t^12-202259378*k^5*t^13 +771249064*k^4*t^14+173628410*k^3*t^15-113461466*k^2*t^16-9005874*k*t^17+ 5278608*t^18-12100988*k^8*t^9+150804670*k^7*t^10-415500479*k^6*t^11-159371902*k ^5*t^12+730656176*k^4*t^13+187861472*k^3*t^14-122493135*k^2*t^15-14996928*k*t^ 16+6066792*t^17-5939904*k^8*t^8+87312184*k^7*t^9-280672389*k^6*t^10-103243622*k ^5*t^11+618697565*k^4*t^12+173628410*k^3*t^13-125155380*k^2*t^14-19547652*k*t^ 15+5437152*t^16-2465960*k^8*t^7+44014740*k^7*t^8-168682089*k^6*t^9-54066059*k^5 *t^10+466849732*k^4*t^11+136834784*k^3*t^12-122493135*k^2*t^13-21295896*k*t^14+ 4204872*t^15-833782*k^8*t^6+18940476*k^7*t^7-89533192*k^6*t^8-18530628*k^5*t^9+ 315455760*k^4*t^10+88428502*k^3*t^11-113461466*k^2*t^12-19547652*k*t^13+3619296 *t^14-217754*k^8*t^5+6731620*k^7*t^6-41106416*k^6*t^7+932652*k^5*t^8+187682134* k^4*t^9+40617409*k^3*t^10-95987629*k^2*t^11-14996928*k*t^12+4204872*t^13-34686* k^8*t^4+1896104*k^7*t^5-15812878*k^6*t^6+6632116*k^5*t^7+95327282*k^4*t^8+ 7362134*k^3*t^9-71080406*k^2*t^10-9005874*k*t^11+5437152*t^12-734*k^8*t^3+ 352196*k^7*t^4-4940453*k^6*t^5+5362702*k^5*t^6+40207746*k^4*t^7-6799140*k^3*t^8 -44583521*k^2*t^9-3171756*k*t^10+6066792*t^11+685*k^8*t^2+25216*k^7*t^3-1050215 *k^6*t^4+2813036*k^5*t^5+13375682*k^4*t^6-8270470*k^3*t^7-23143306*k^2*t^8+ 685086*k*t^9+5278608*t^10+90*k^8*t-2758*k^7*t^2-104586*k^6*t^3+823207*k^5*t^4+ 2998288*k^4*t^5-4948970*k^3*t^6-9613754*k^2*t^7+1993332*k*t^8+3530952*t^9+3*k^8 -498*k^7*t+1995*k^6*t^2+95619*k^5*t^3+409367*k^4*t^4-1784710*k^3*t^5-3046182*k^ 2*t^6+1676670*k*t^7+1812240*t^8-14*k^7+874*k^6*t-1507*k^5*t^2+55849*k^4*t^3-\ 400719*k^3*t^4-690561*k^2*t^5+889584*k*t^6+680616*t^7+5*k^6-711*k^5*t+7650*k^4* t^2-67465*k^3*t^3-101442*k^2*t^4+297066*k*t^5+165456*t^6+59*k^5+571*k^4*t-6511* k^3*t^2-4193*k^2*t^3+54708*k*t^4+20952*t^5-110*k^4-513*k^3*t+686*k^2*t^2+3966*k *t^3+864*t^4+75*k^3+217*k^2*t-240*k*t^2-216*t^3-18*k^2-30*k*t)/(k+1)^2/(2*k+1)^ 2/(k^6*t^18+34*k^6*t^17-7*k^5*t^18+384*k^6*t^16-276*k^5*t^17+15*k^4*t^18+2010*k ^6*t^15-3152*k^5*t^16+851*k^4*t^17-3*k^3*t^18+5818*k^6*t^14-15844*k^5*t^15+9940 *k^4*t^16-1274*k^3*t^17-28*k^2*t^18+12520*k^6*t^13-43046*k^5*t^14+47910*k^4*t^ 15-15492*k^3*t^16+975*k^2*t^17+34*k*t^18+24716*k^6*t^12-88466*k^5*t^13+119016*k ^4*t^14-71669*k^3*t^15+12644*k^2*t^16-358*k*t^17-12*t^18+36422*k^6*t^11-175548* k^5*t^12+229971*k^4*t^13-156554*k^3*t^14+56334*k^2*t^15-5164*k*t^16+48*t^17+ 47737*k^6*t^10-251540*k^5*t^11+467439*k^4*t^12-280283*k^3*t^13+100646*k^2*t^14-\ 22443*k*t^15+840*t^16+51756*k^6*t^9-329527*k^5*t^10+640658*k^4*t^11-610712*k^3* t^12+164279*k^2*t^13-28184*k*t^14+3678*t^15+47737*k^6*t^8-357284*k^5*t^9+842230 *k^4*t^10-779119*k^3*t^11+423799*k^2*t^12-41135*k*t^13+2160*t^14+36422*k^6*t^7-\ 329527*k^5*t^8+915492*k^4*t^9-1037075*k^3*t^10+475622*k^2*t^11-155446*k*t^12+ 2466*t^13+24716*k^6*t^6-251540*k^5*t^7+842230*k^4*t^8-1138246*k^3*t^9+650359*k^ 2*t^10-136341*k*t^11+24672*t^12+12520*k^6*t^5-175548*k^5*t^6+640658*k^4*t^7-\ 1037075*k^3*t^8+732612*k^2*t^9-195168*k*t^10+13650*t^11+5818*k^6*t^4-88466*k^5* t^5+467439*k^4*t^6-779119*k^3*t^7+650359*k^2*t^8-233862*k*t^9+21300*t^10+2010*k ^6*t^3-43046*k^5*t^4+229971*k^4*t^5-610712*k^3*t^6+475622*k^2*t^7-195168*k*t^8+ 29436*t^9+384*k^6*t^2-15844*k^5*t^3+119016*k^4*t^4-280283*k^3*t^5+423799*k^2*t^ 6-136341*k*t^7+21300*t^8+34*k^6*t-3152*k^5*t^2+47910*k^4*t^3-156554*k^3*t^4+ 164279*k^2*t^5-155446*k*t^6+13650*t^7+k^6-276*k^5*t+9940*k^4*t^2-71669*k^3*t^3+ 100646*k^2*t^4-41135*k*t^5+24672*t^6-7*k^5+851*k^4*t-15492*k^3*t^2+56334*k^2*t^ 3-28184*k*t^4+2466*t^5+15*k^4-1274*k^3*t+12644*k^2*t^2-22443*k*t^3+2160*t^4-3*k ^3+975*k^2*t-5164*k*t^2+3678*t^3-28*k^2-358*k*t+840*t^2+34*k+48*t-12)/k/t/K^3+1 /4*(t^2+t+1)^3*(t+1)^6*(t-1)^14*(k-1)*(k-2)^2*(k-3)^2*(k^6*t^18+34*k^6*t^17-k^5 *t^18+384*k^6*t^16-72*k^5*t^17-5*k^4*t^18+2010*k^6*t^15-848*k^5*t^16-19*k^4*t^ 17+7*k^3*t^18+5818*k^6*t^14-3784*k^5*t^15-60*k^4*t^16+50*k^3*t^17-2*k^2*t^18+ 12520*k^6*t^13-8138*k^5*t^14-1160*k^4*t^15+428*k^3*t^16+9*k^2*t^17+24716*k^6*t^ 12-13346*k^5*t^13-8944*k^4*t^14+1731*k^3*t^15+48*k^2*t^16-2*k*t^17+36422*k^6*t^ 11-27252*k^5*t^12-24559*k^4*t^13+5410*k^3*t^14+497*k^2*t^15-48*k*t^16+47737*k^6 *t^10-33008*k^5*t^11-39561*k^4*t^12+5341*k^3*t^13+1890*k^2*t^14-302*k*t^15+ 51756*k^6*t^9-43105*k^5*t^10-70712*k^4*t^11-2116*k^3*t^12+6396*k^2*t^13-812*k*t ^14-24*t^15+47737*k^6*t^8-46748*k^5*t^9-89350*k^4*t^10-3447*k^3*t^11+11557*k^2* t^12-752*k*t^13-144*t^14+36422*k^6*t^7-43105*k^5*t^8-94588*k^4*t^9-8685*k^3*t^ 10+13143*k^2*t^11+328*k*t^12-648*t^13+24716*k^6*t^6-33008*k^5*t^7-89350*k^4*t^8 -13998*k^3*t^9+13299*k^2*t^10+1010*k*t^11-1080*t^12+12520*k^6*t^5-27252*k^5*t^6 -70712*k^4*t^7-8685*k^3*t^8+14326*k^2*t^9+2032*k*t^10-648*t^11+5818*k^6*t^4-\ 13346*k^5*t^5-39561*k^4*t^6-3447*k^3*t^7+13299*k^2*t^8+2708*k*t^9-144*t^10+2010 *k^6*t^3-8138*k^5*t^4-24559*k^4*t^5-2116*k^3*t^6+13143*k^2*t^7+2032*k*t^8-96*t^ 9+384*k^6*t^2-3784*k^5*t^3-8944*k^4*t^4+5341*k^3*t^5+11557*k^2*t^6+1010*k*t^7-\ 144*t^8+34*k^6*t-848*k^5*t^2-1160*k^4*t^3+5410*k^3*t^4+6396*k^2*t^5+328*k*t^6-\ 648*t^7+k^6-72*k^5*t-60*k^4*t^2+1731*k^3*t^3+1890*k^2*t^4-752*k*t^5-1080*t^6-k^ 5-19*k^4*t+428*k^3*t^2+497*k^2*t^3-812*k*t^4-648*t^5-5*k^4+50*k^3*t+48*k^2*t^2-\ 302*k*t^3-144*t^4+7*k^3+9*k^2*t-48*k*t^2-24*t^3-2*k^2-2*k*t)/(k+1)^2/(2*k+1)^2/ (k^6*t^18+34*k^6*t^17-7*k^5*t^18+384*k^6*t^16-276*k^5*t^17+15*k^4*t^18+2010*k^6 *t^15-3152*k^5*t^16+851*k^4*t^17-3*k^3*t^18+5818*k^6*t^14-15844*k^5*t^15+9940*k ^4*t^16-1274*k^3*t^17-28*k^2*t^18+12520*k^6*t^13-43046*k^5*t^14+47910*k^4*t^15-\ 15492*k^3*t^16+975*k^2*t^17+34*k*t^18+24716*k^6*t^12-88466*k^5*t^13+119016*k^4* t^14-71669*k^3*t^15+12644*k^2*t^16-358*k*t^17-12*t^18+36422*k^6*t^11-175548*k^5 *t^12+229971*k^4*t^13-156554*k^3*t^14+56334*k^2*t^15-5164*k*t^16+48*t^17+47737* k^6*t^10-251540*k^5*t^11+467439*k^4*t^12-280283*k^3*t^13+100646*k^2*t^14-22443* k*t^15+840*t^16+51756*k^6*t^9-329527*k^5*t^10+640658*k^4*t^11-610712*k^3*t^12+ 164279*k^2*t^13-28184*k*t^14+3678*t^15+47737*k^6*t^8-357284*k^5*t^9+842230*k^4* t^10-779119*k^3*t^11+423799*k^2*t^12-41135*k*t^13+2160*t^14+36422*k^6*t^7-\ 329527*k^5*t^8+915492*k^4*t^9-1037075*k^3*t^10+475622*k^2*t^11-155446*k*t^12+ 2466*t^13+24716*k^6*t^6-251540*k^5*t^7+842230*k^4*t^8-1138246*k^3*t^9+650359*k^ 2*t^10-136341*k*t^11+24672*t^12+12520*k^6*t^5-175548*k^5*t^6+640658*k^4*t^7-\ 1037075*k^3*t^8+732612*k^2*t^9-195168*k*t^10+13650*t^11+5818*k^6*t^4-88466*k^5* t^5+467439*k^4*t^6-779119*k^3*t^7+650359*k^2*t^8-233862*k*t^9+21300*t^10+2010*k ^6*t^3-43046*k^5*t^4+229971*k^4*t^5-610712*k^3*t^6+475622*k^2*t^7-195168*k*t^8+ 29436*t^9+384*k^6*t^2-15844*k^5*t^3+119016*k^4*t^4-280283*k^3*t^5+423799*k^2*t^ 6-136341*k*t^7+21300*t^8+34*k^6*t-3152*k^5*t^2+47910*k^4*t^3-156554*k^3*t^4+ 164279*k^2*t^5-155446*k*t^6+13650*t^7+k^6-276*k^5*t+9940*k^4*t^2-71669*k^3*t^3+ 100646*k^2*t^4-41135*k*t^5+24672*t^6-7*k^5+851*k^4*t-15492*k^3*t^2+56334*k^2*t^ 3-28184*k*t^4+2466*t^5+15*k^4-1274*k^3*t+12644*k^2*t^2-22443*k*t^3+2160*t^4-3*k ^3+975*k^2*t-5164*k*t^2+3678*t^3-28*k^2-358*k*t+840*t^2+34*k+48*t-12)/k/t/K^4: RETURN(normal(add(subs(k=k1,coeff(ope,K,-i))*Bp3(k1-i,t),i=1..4))): fi: end: #Bp2(k,t): A fast way to compute Bp(2,k,t) Bp2:=proc(k,t) option remember: if k=0 then RETURN(2): elif k=1 then 2*t^2+2*t+2: elif k=2 then RETURN(2*t^4+4*t^3+8*t^2+4*t+2): else expand(2*(k*t^2+k+t)/(k+1)*Bp2(k-1,t)-(t-1)^2*(t+1)^2*(k-1)/(k+1)*Bp2(k-2,t)): fi: end: Bn3:=proc(k1,t) local k,K,ope,i,INI: option remember: ope:= (3*k^8*t^28+90*k^8*t^27-82*k^7*t^28+685*k^8*t^26-2382*k^7*t^27+957*k^6*t^28-734 *k^8*t^25-19162*k^7*t^26+27250*k^6*t^27-6227*k^5*t^28-34686*k^8*t^24-1728*k^7*t ^25+231651*k^6*t^26-175497*k^5*t^27+24670*k^4*t^28-217754*k^8*t^23+757756*k^7*t ^24+272630*k^6*t^25-1574725*k^5*t^26+693391*k^4*t^27-60747*k^3*t^28-833782*k^8* t^22+5072024*k^7*t^23-6728055*k^6*t^24-3427475*k^5*t^25+6553590*k^4*t^26-\ 1712463*k^3*t^27+90290*k^2*t^28-2465960*k^8*t^21+19949404*k^7*t^22-49403333*k^6 *t^23+30358721*k^5*t^24+20198149*k^4*t^25-16997329*k^3*t^26+2562925*k^2*t^27-\ 73344*k*t^28-5939904*k^8*t^20+59970244*k^7*t^21-200861854*k^6*t^22+259097228*k^ 5*t^23-67832173*k^4*t^24-66099943*k^3*t^25+26606522*k^2*t^26-2100858*k*t^27+ 24480*t^28-12100988*k^8*t^19+146062188*k^7*t^20-615523168*k^6*t^21+1100596738*k ^5*t^22-781328432*k^4*t^23+39770751*k^3*t^24+122197523*k^2*t^25-22686384*k*t^26 +707544*t^27-21299821*k^8*t^18+299919432*k^7*t^19-1518197464*k^6*t^20+ 3453922572*k^5*t^21-3537005638*k^4*t^22+1325299718*k^3*t^23+123970858*k^2*t^24-\ 117631686*k*t^25+7885152*t^26-32799928*k^8*t^17+530789602*k^7*t^18-3145183561*k ^6*t^19+8647527252*k^5*t^20-11456026734*k^4*t^21+6670469386*k^3*t^22-1105218249 *k^2*t^23-237790932*k*t^24+44493264*t^25-44406725*k^8*t^16+820431334*k^7*t^17-\ 5600461437*k^6*t^18+18099947932*k^5*t^19-29224047838*k^4*t^20+22621365734*k^3*t ^21-6935724126*k^2*t^22+253147326*k*t^23+116229456*t^24-53206022*k^8*t^15+ 1113554282*k^7*t^16-8693210087*k^6*t^17+32458392739*k^5*t^18-61937044586*k^4*t^ 19+59180608644*k^3*t^20-25373023554*k^2*t^21+3403385376*k*t^22+82995696*t^23-\ 56496788*k^8*t^14+1336070020*k^7*t^15-11832368878*k^6*t^16+50630299438*k^5*t^17 -112017570300*k^4*t^18+127469585354*k^3*t^19-68906614906*k^2*t^20+14404725618*k *t^21-567591120*t^22-53206022*k^8*t^13+1419308600*k^7*t^14-14219189267*k^6*t^15 +69135688622*k^5*t^16-175760196548*k^4*t^17+233030410015*k^3*t^18-151814483737* k^2*t^19+41568336540*k*t^20-3251548800*t^21-44406725*k^8*t^12+1336070020*k^7*t^ 13-15112403600*k^6*t^14+83233657914*k^5*t^15-240919623835*k^4*t^16+368350036138 *k^3*t^17-281614417226*k^2*t^18+94743692490*k*t^19-10304835264*t^20-32799928*k^ 8*t^11+1113554282*k^7*t^12-14219189267*k^6*t^13+88512319328*k^5*t^14-\ 290682008584*k^4*t^15+507312002576*k^3*t^16-449579722789*k^2*t^17+179459858220* k*t^18-24613988688*t^19-21299821*k^8*t^10+820431334*k^7*t^11-11832368878*k^6*t^ 12+83233657914*k^5*t^13-309327096536*k^4*t^14+613774250822*k^3*t^15-\ 623070634106*k^2*t^16+290499818106*k*t^17-47907611712*t^18-12100988*k^8*t^9+ 530789602*k^7*t^10-8693210087*k^6*t^11+69135688622*k^5*t^12-290682008584*k^4*t^ 13+653697953696*k^3*t^14-756552417559*k^2*t^15+406070483856*k*t^16-78918640560* t^17-5939904*k^8*t^8+299919432*k^7*t^9-5600461437*k^6*t^10+50630299438*k^5*t^11 -240919623835*k^4*t^12+613774250822*k^3*t^13-806662390452*k^2*t^14+495505874844 *k*t^15-111487267008*t^16-2465960*k^8*t^7+146062188*k^7*t^8-3145183561*k^6*t^9+ 32458392739*k^5*t^10-175760196548*k^4*t^11+507312002576*k^3*t^12-756552417559*k ^2*t^13+529125151416*k*t^14-136866284856*t^15-833782*k^8*t^6+59970244*k^7*t^7-\ 1518197464*k^6*t^8+18099947932*k^5*t^9-112017570300*k^4*t^10+368350036138*k^3*t ^11-623070634106*k^2*t^12+495505874844*k*t^13-146419512768*t^14-217754*k^8*t^5+ 19949404*k^7*t^6-615523168*k^6*t^7+8647527252*k^5*t^8-61937044586*k^4*t^9+ 233030410015*k^3*t^10-449579722789*k^2*t^11+406070483856*k*t^12-136866284856*t^ 13-34686*k^8*t^4+5072024*k^7*t^5-200861854*k^6*t^6+3453922572*k^5*t^7-\ 29224047838*k^4*t^8+127469585354*k^3*t^9-281614417226*k^2*t^10+290499818106*k*t ^11-111487267008*t^12-734*k^8*t^3+757756*k^7*t^4-49403333*k^6*t^5+1100596738*k^ 5*t^6-11456026734*k^4*t^7+59180608644*k^3*t^8-151814483737*k^2*t^9+179459858220 *k*t^10-78918640560*t^11+685*k^8*t^2-1728*k^7*t^3-6728055*k^6*t^4+259097228*k^5 *t^5-3537005638*k^4*t^6+22621365734*k^3*t^7-68906614906*k^2*t^8+94743692490*k*t ^9-47907611712*t^10+90*k^8*t-19162*k^7*t^2+272630*k^6*t^3+30358721*k^5*t^4-\ 781328432*k^4*t^5+6670469386*k^3*t^6-25373023554*k^2*t^7+41568336540*k*t^8-\ 24613988688*t^9+3*k^8-2382*k^7*t+231651*k^6*t^2-3427475*k^5*t^3-67832173*k^4*t^ 4+1325299718*k^3*t^5-6935724126*k^2*t^6+14404725618*k*t^7-10304835264*t^8-82*k^ 7+27250*k^6*t-1574725*k^5*t^2+20198149*k^4*t^3+39770751*k^3*t^4-1105218249*k^2* t^5+3403385376*k*t^6-3251548800*t^7+957*k^6-175497*k^5*t+6553590*k^4*t^2-\ 66099943*k^3*t^3+123970858*k^2*t^4+253147326*k*t^5-567591120*t^6-6227*k^5+ 693391*k^4*t-16997329*k^3*t^2+122197523*k^2*t^3-237790932*k*t^4+82995696*t^5+ 24670*k^4-1712463*k^3*t+26606522*k^2*t^2-117631686*k*t^3+116229456*t^4-60747*k^ 3+2562925*k^2*t-22686384*k*t^2+44493264*t^3+90290*k^2-2100858*k*t+7885152*t^2-\ 73344*k+707544*t+24480)/(t+1)^4/(t-1)^6/(t^2+t+1)^3/(k-1)^2/(k^6*t^18+34*k^6*t^ 17-23*k^5*t^18+384*k^6*t^16-744*k^5*t^17+215*k^4*t^18+2010*k^6*t^15-8368*k^5*t^ 16+6701*k^4*t^17-1047*k^3*t^18+5818*k^6*t^14-44456*k^5*t^15+75140*k^4*t^16-\ 31746*k^3*t^17+2802*k^2*t^18+12520*k^6*t^13-131494*k^5*t^14+405560*k^4*t^15-\ 355308*k^3*t^16+83265*k^2*t^17-3904*k*t^18+24716*k^6*t^12-287134*k^5*t^13+ 1224616*k^4*t^14-1950531*k^3*t^15+931264*k^2*t^16-114342*k*t^17+2208*t^18+36422 *k^6*t^11-565932*k^5*t^12+2713321*k^4*t^13-6007266*k^3*t^14+5206549*k^2*t^15-\ 1279376*k*t^16+64008*t^17+47737*k^6*t^10-841120*k^5*t^11+5347239*k^4*t^12-\ 13502637*k^3*t^13+16340986*k^2*t^14-7295722*k*t^15+717120*t^16+51756*k^6*t^9-\ 1102583*k^5*t^10+8010408*k^4*t^11-26641068*k^3*t^12+37248184*k^2*t^13-23313476* k*t^14+4178688*t^15+47737*k^6*t^8-1195396*k^5*t^9+10505430*k^4*t^10-40204041*k^ 3*t^11+73656469*k^2*t^12-53859680*k*t^13+13580640*t^14+36422*k^6*t^7-1102583*k^ 5*t^8+11391892*k^4*t^9-52768275*k^3*t^10+111918787*k^2*t^11-106836144*k*t^12+ 31769016*t^13+24716*k^6*t^6-841120*k^5*t^7+10505430*k^4*t^8-57240594*k^3*t^9+ 147054359*k^2*t^10-163364954*k*t^11+63252792*t^12+12520*k^6*t^5-565932*k^5*t^6+ 8010408*k^4*t^7-52768275*k^3*t^8+159590222*k^2*t^9-214939672*k*t^10+97275120*t^ 11+5818*k^6*t^4-287134*k^5*t^5+5347239*k^4*t^6-40204041*k^3*t^7+147054359*k^2*t ^8-233382308*k*t^9+128182560*t^10+2010*k^6*t^3-131494*k^5*t^4+2713321*k^4*t^5-\ 26641068*k^3*t^6+111918787*k^2*t^7-214939672*k*t^8+139252176*t^9+384*k^6*t^2-\ 44456*k^5*t^3+1224616*k^4*t^4-13502637*k^3*t^5+73656469*k^2*t^6-163364954*k*t^7 +128182560*t^8+34*k^6*t-8368*k^5*t^2+405560*k^4*t^3-6007266*k^3*t^4+37248184*k^ 2*t^5-106836144*k*t^6+97275120*t^7+k^6-744*k^5*t+75140*k^4*t^2-1950531*k^3*t^3+ 16340986*k^2*t^4-53859680*k*t^5+63252792*t^6-23*k^5+6701*k^4*t-355308*k^3*t^2+ 5206549*k^2*t^3-23313476*k*t^4+31769016*t^5+215*k^4-31746*k^3*t+931264*k^2*t^2-\ 7295722*k*t^3+13580640*t^4-1047*k^3+83265*k^2*t-1279376*k*t^2+4178688*t^3+2802* k^2-114342*k*t+717120*t^2-3904*k+64008*t+2208)/K-(3*k^10*t^30+47*k^10*t^29-101* k^9*t^30-763*k^10*t^28-1547*k^9*t^29+1505*k^8*t^30-17000*k^10*t^27+22511*k^9*t^ 28+22822*k^8*t^29-13065*k^7*t^30-124197*k^10*t^26+527076*k^9*t^27-293255*k^8*t^ 28-198018*k^7*t^29+73110*k^6*t^30-550791*k^10*t^25+3979177*k^9*t^26-7266457*k^8 *t^27+2217612*k^7*t^28+1115235*k^6*t^29-275190*k^5*t^30-2033626*k^10*t^24+ 18140711*k^9*t^25-56741166*k^8*t^26+58608678*k^7*t^27-10758705*k^6*t^28-4245423 *k^5*t^29+704262*k^4*t^30-6406837*k^10*t^23+67756442*k^9*t^24-265750545*k^8*t^ 25+473806953*k^7*t^26-305943690*k^6*t^27+34906923*k^5*t^28+11022712*k^4*t^29-\ 1206700*k^3*t^30-16300174*k^10*t^22+214244865*k^9*t^23-1004745691*k^8*t^24+ 2278444488*k^7*t^25-2563081812*k^6*t^26+1078579164*k^5*t^27-76491017*k^4*t^28-\ 19195012*k^3*t^29+1319632*k^2*t^30-36127405*k^10*t^21+549041424*k^9*t^22-\ 3191255129*k^8*t^23+8724324921*k^7*t^24-12647765331*k^6*t^25+9373041192*k^5*t^ 26-2596127557*k^4*t^27+111409858*k^3*t^28+21347408*k^2*t^29-826944*k*t^30-\ 69426725*k^10*t^20+1221334153*k^9*t^21-8236611974*k^8*t^22+27855590115*k^7*t^23 -49068058065*k^6*t^24+47434687833*k^5*t^25-23426030577*k^4*t^26+4202775642*k^3* t^27-102834772*k^2*t^28-13598400*k*t^29+223488*t^30-117248564*k^10*t^19+ 2353331731*k^9*t^20-18392578626*k^8*t^21+72404195799*k^7*t^22-157599319752*k^6* t^23+186512960067*k^5*t^24-121510278441*k^4*t^25+39419174198*k^3*t^26-\ 4364634800*k^2*t^27+54105096*k*t^28+3730176*t^29-174835275*k^10*t^18+3982462208 *k^9*t^19-35540947501*k^8*t^20+162331021683*k^7*t^21-412535487093*k^6*t^22+ 602987051514*k^5*t^23-484334587058*k^4*t^24+209422828504*k^3*t^25-42599826792*k 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t-279371338*k^4*t^2+6526507641*k^3*t^3-49365009008*k^2*t^4+131787311124*k*t^5-\ 118423379232*t^6-110175*k^5-3969141*k^4*t+474663176*k^3*t^2-7152418066*k^2*t^3+ 31911051360*k*t^4-37478235600*t^5+290648*k^4+6693014*k^3*t-516153804*k^2*t^2+ 4500912096*k*t^3-8938237536*t^4-511786*k^3-7201688*k^2*t+322344600*k*t^2-\ 1222999200*t^3+573292*k^2+4442016*k*t-86963616*t^2-366720*k-1183104*t+100800)/( t^2+t+1)^3/(t+1)^6/(t-1)^14/(k-3)/(k-2)^2/(k-1)^2/(k^6*t^18+34*k^6*t^17-23*k^5* t^18+384*k^6*t^16-744*k^5*t^17+215*k^4*t^18+2010*k^6*t^15-8368*k^5*t^16+6701*k^ 4*t^17-1047*k^3*t^18+5818*k^6*t^14-44456*k^5*t^15+75140*k^4*t^16-31746*k^3*t^17 +2802*k^2*t^18+12520*k^6*t^13-131494*k^5*t^14+405560*k^4*t^15-355308*k^3*t^16+ 83265*k^2*t^17-3904*k*t^18+24716*k^6*t^12-287134*k^5*t^13+1224616*k^4*t^14-\ 1950531*k^3*t^15+931264*k^2*t^16-114342*k*t^17+2208*t^18+36422*k^6*t^11-565932* k^5*t^12+2713321*k^4*t^13-6007266*k^3*t^14+5206549*k^2*t^15-1279376*k*t^16+ 64008*t^17+47737*k^6*t^10-841120*k^5*t^11+5347239*k^4*t^12-13502637*k^3*t^13+ 16340986*k^2*t^14-7295722*k*t^15+717120*t^16+51756*k^6*t^9-1102583*k^5*t^10+ 8010408*k^4*t^11-26641068*k^3*t^12+37248184*k^2*t^13-23313476*k*t^14+4178688*t^ 15+47737*k^6*t^8-1195396*k^5*t^9+10505430*k^4*t^10-40204041*k^3*t^11+73656469*k ^2*t^12-53859680*k*t^13+13580640*t^14+36422*k^6*t^7-1102583*k^5*t^8+11391892*k^ 4*t^9-52768275*k^3*t^10+111918787*k^2*t^11-106836144*k*t^12+31769016*t^13+24716 *k^6*t^6-841120*k^5*t^7+10505430*k^4*t^8-57240594*k^3*t^9+147054359*k^2*t^10-\ 163364954*k*t^11+63252792*t^12+12520*k^6*t^5-565932*k^5*t^6+8010408*k^4*t^7-\ 52768275*k^3*t^8+159590222*k^2*t^9-214939672*k*t^10+97275120*t^11+5818*k^6*t^4-\ 287134*k^5*t^5+5347239*k^4*t^6-40204041*k^3*t^7+147054359*k^2*t^8-233382308*k*t ^9+128182560*t^10+2010*k^6*t^3-131494*k^5*t^4+2713321*k^4*t^5-26641068*k^3*t^6+ 111918787*k^2*t^7-214939672*k*t^8+139252176*t^9+384*k^6*t^2-44456*k^5*t^3+ 1224616*k^4*t^4-13502637*k^3*t^5+73656469*k^2*t^6-163364954*k*t^7+128182560*t^8 +34*k^6*t-8368*k^5*t^2+405560*k^4*t^3-6007266*k^3*t^4+37248184*k^2*t^5-\ 106836144*k*t^6+97275120*t^7+k^6-744*k^5*t+75140*k^4*t^2-1950531*k^3*t^3+ 16340986*k^2*t^4-53859680*k*t^5+63252792*t^6-23*k^5+6701*k^4*t-355308*k^3*t^2+ 5206549*k^2*t^3-23313476*k*t^4+31769016*t^5+215*k^4-31746*k^3*t+931264*k^2*t^2-\ 7295722*k*t^3+13580640*t^4-1047*k^3+83265*k^2*t-1279376*k*t^2+4178688*t^3+2802* k^2-114342*k*t+717120*t^2-3904*k+64008*t+2208)/K^3+4*t*(k-4)*(k-5)^2*(2*k-9)^2* (k^6*t^18+34*k^6*t^17-17*k^5*t^18+384*k^6*t^16-540*k^5*t^17+115*k^4*t^18+2010*k ^6*t^15-6064*k^5*t^16+3491*k^4*t^17-397*k^3*t^18+5818*k^6*t^14-32396*k^5*t^15+ 39060*k^4*t^16-11702*k^3*t^17+736*k^2*t^18+12520*k^6*t^13-96586*k^5*t^14+213430 *k^4*t^15-130748*k^3*t^16+21303*k^2*t^17-690*k*t^18+24716*k^6*t^12-212014*k^5*t ^13+654416*k^4*t^14-732651*k^3*t^15+238260*k^2*t^16-19762*k*t^17+252*t^18+36422 *k^6*t^11-417636*k^5*t^12+1465451*k^4*t^13-2307382*k^3*t^14+1373906*k^2*t^15-\ 221748*k*t^16+7176*t^17+47737*k^6*t^10-622588*k^5*t^11+2888319*k^4*t^12-5270293 *k^3*t^13+4439214*k^2*t^14-1322197*k*t^15+80856*t^16+51756*k^6*t^9-816161*k^5*t ^10+4351138*k^4*t^11-10417112*k^3*t^12+10336659*k^2*t^13-4377400*k*t^14+502098* t^15+47737*k^6*t^8-884860*k^5*t^9+5708570*k^4*t^10-15845169*k^3*t^11+20528119*k ^2*t^12-10378489*k*t^13+1699824*t^14+36422*k^6*t^7-816161*k^5*t^8+6191252*k^4*t ^9-20817645*k^3*t^10+31504242*k^2*t^11-20738818*k*t^12+4093590*t^13+24716*k^6*t ^6-622588*k^5*t^7+5708570*k^4*t^8-22591866*k^3*t^9+41472339*k^2*t^10-32084939*k *t^11+8238072*t^12+12520*k^6*t^5-417636*k^5*t^6+4351138*k^4*t^7-20817645*k^3*t^ 8+45042172*k^2*t^9-42340552*k*t^10+12830622*t^11+5818*k^6*t^4-212014*k^5*t^5+ 2888319*k^4*t^6-15845169*k^3*t^7+41472339*k^2*t^8-46022522*k*t^9+16979556*t^10+ 2010*k^6*t^3-96586*k^5*t^4+1465451*k^4*t^5-10417112*k^3*t^6+31504242*k^2*t^7-\ 42340552*k*t^8+18467748*t^9+384*k^6*t^2-32396*k^5*t^3+654416*k^4*t^4-5270293*k^ 3*t^5+20528119*k^2*t^6-32084939*k*t^7+16979556*t^8+34*k^6*t-6064*k^5*t^2+213430 *k^4*t^3-2307382*k^3*t^4+10336659*k^2*t^5-20738818*k*t^6+12830622*t^7+k^6-540*k ^5*t+39060*k^4*t^2-732651*k^3*t^3+4439214*k^2*t^4-10378489*k*t^5+8238072*t^6-17 *k^5+3491*k^4*t-130748*k^3*t^2+1373906*k^2*t^3-4377400*k*t^4+4093590*t^5+115*k^ 4-11702*k^3*t+238260*k^2*t^2-1322197*k*t^3+1699824*t^4-397*k^3+21303*k^2*t-\ 221748*k*t^2+502098*t^3+736*k^2-19762*k*t+80856*t^2-690*k+7176*t+252)/(t^2+t+1) ^3/(t+1)^6/(t-1)^14/(k-3)/(k-2)^2/(k-1)^2/(k^6*t^18+34*k^6*t^17-23*k^5*t^18+384 *k^6*t^16-744*k^5*t^17+215*k^4*t^18+2010*k^6*t^15-8368*k^5*t^16+6701*k^4*t^17-\ 1047*k^3*t^18+5818*k^6*t^14-44456*k^5*t^15+75140*k^4*t^16-31746*k^3*t^17+2802*k ^2*t^18+12520*k^6*t^13-131494*k^5*t^14+405560*k^4*t^15-355308*k^3*t^16+83265*k^ 2*t^17-3904*k*t^18+24716*k^6*t^12-287134*k^5*t^13+1224616*k^4*t^14-1950531*k^3* t^15+931264*k^2*t^16-114342*k*t^17+2208*t^18+36422*k^6*t^11-565932*k^5*t^12+ 2713321*k^4*t^13-6007266*k^3*t^14+5206549*k^2*t^15-1279376*k*t^16+64008*t^17+ 47737*k^6*t^10-841120*k^5*t^11+5347239*k^4*t^12-13502637*k^3*t^13+16340986*k^2* t^14-7295722*k*t^15+717120*t^16+51756*k^6*t^9-1102583*k^5*t^10+8010408*k^4*t^11 -26641068*k^3*t^12+37248184*k^2*t^13-23313476*k*t^14+4178688*t^15+47737*k^6*t^8 -1195396*k^5*t^9+10505430*k^4*t^10-40204041*k^3*t^11+73656469*k^2*t^12-53859680 *k*t^13+13580640*t^14+36422*k^6*t^7-1102583*k^5*t^8+11391892*k^4*t^9-52768275*k ^3*t^10+111918787*k^2*t^11-106836144*k*t^12+31769016*t^13+24716*k^6*t^6-841120* k^5*t^7+10505430*k^4*t^8-57240594*k^3*t^9+147054359*k^2*t^10-163364954*k*t^11+ 63252792*t^12+12520*k^6*t^5-565932*k^5*t^6+8010408*k^4*t^7-52768275*k^3*t^8+ 159590222*k^2*t^9-214939672*k*t^10+97275120*t^11+5818*k^6*t^4-287134*k^5*t^5+ 5347239*k^4*t^6-40204041*k^3*t^7+147054359*k^2*t^8-233382308*k*t^9+128182560*t^ 10+2010*k^6*t^3-131494*k^5*t^4+2713321*k^4*t^5-26641068*k^3*t^6+111918787*k^2*t ^7-214939672*k*t^8+139252176*t^9+384*k^6*t^2-44456*k^5*t^3+1224616*k^4*t^4-\ 13502637*k^3*t^5+73656469*k^2*t^6-163364954*k*t^7+128182560*t^8+34*k^6*t-8368*k ^5*t^2+405560*k^4*t^3-6007266*k^3*t^4+37248184*k^2*t^5-106836144*k*t^6+97275120 *t^7+k^6-744*k^5*t+75140*k^4*t^2-1950531*k^3*t^3+16340986*k^2*t^4-53859680*k*t^ 5+63252792*t^6-23*k^5+6701*k^4*t-355308*k^3*t^2+5206549*k^2*t^3-23313476*k*t^4+ 31769016*t^5+215*k^4-31746*k^3*t+931264*k^2*t^2-7295722*k*t^3+13580640*t^4-1047 *k^3+83265*k^2*t-1279376*k*t^2+4178688*t^3+2802*k^2-114342*k*t+717120*t^2-3904* k+64008*t+2208)/K^4: INI:= [1, 1/(t+1)/(t^2+t+1), (t^4-t^2+1)/(t-1)^4/(t+1)^4/(t^2+t+1)^4, (t^14-t^13-\ 2*t^12+6*t^11+6*t^10-9*t^9+t^8+17*t^7+t^6-9*t^5+6*t^4+6*t^3-2*t^2-t+1)/(t^2+t+1 )^7/(t+1)^8/(t-1)^10]: if k1<=3 then RETURN(INI[k1+1]): else RETURN(normal(add(subs(k=k1,coeff(ope,K,-i))*Bn3(k1-i,t),i=1..4))): fi: end: #CheckBn3(k,N): Checks that the first N Taylor coefficients of Bn3(k,t) are the same as those of B(n,3,t) CheckBn3:=proc(k,N) local gu1,gu2,gu,t,k1,i: for k1 from 1 to k do gu:=Bn3(k,t): gu:=taylor(gu,t=0,N+3): gu1:=[seq(coeff(gu,t,i),i=0..N)]: gu:=BnTaylor(3,k,t,N+4): gu2:=[seq(coeff(gu,t,i),i=0..N)]: if gu1<>gu2 then print(k1, `did not work out`): RETURN(false): fi: od: true: end: #Cnk(n,k,t): The Poincare series of Allan Berele's C bar(n,k) in the variable t, done directly. For large n and k it is very slow. Cnk:=proc(n,k,t) : 1/n!*normal(Bn(n,k,t)/(1-t)^(n*k)): end: #C2k(k,t): The Poincare series of Allan Berele's C bar(2,k) in the variable t, done via the recurrence. Much faster. Try: #C2(10,t); C2k:=proc(k,t) : normal(Bn2(k,t)/(1-t)^(2*k)): end: #C3k(k,t): The Poincare series of Allan Berele's C bar(3,k) in the variable t, done via the recurrence. Much faster. Try: #C3(10,t); C3k:=proc(k,t) : normal(Bn3(k,t)/(1-t)^(3*k)): end: #Paper2(K): outputs a paper with the recurrence for the Poincare series of Cbar(2,k) and the first K terms. Try: #Paper2(30); Paper2:=proc(K) local gu,z,k,B2,k1,t: print(``): print(`A recurrence for the Poincare series of Allan Berele's Cbar(2,k) obtained via the Almkvist-Zeilberger algorithm and using it to compute the first`, K, `terms `): print(``): print(`By Shalosh B. Ekhad `): print(``): print(`Thanks to the Almkvist-Zeilberger algorithm we have the following `): print(``): print(`Theorem: Let C2(k,t) be the Poincare series of Cbar(2,k) that according to Allan Berele equals 1/(1-t)^(2*k) times B2(k,t), where B2(k,t) equals `): print(`the constant term, in z of `): print((1-z)*(1-1/z)*((1-t/z)*(1-t*z))^(-k)): print(``): print(`One can compute many terms for B2(k,t) and hence C2(k,t) using the following second-order linear recurrence `): print(``): print(B2(k,t)=2*(k*t^2-2*t^2+k-t-2)/(t-1)^2/(t+1)^2/(k-1)*B2(k-1,t)-(-3+k)/(t-1)^2/(t+1)^2/(k-1)*B2(k-2,t)): print(``): print(`subject to the initial conditions`): print(``): print(B2(0,t)=1, B2(1,t)=1/(1+t)): print(``): print(`using this recurrence, the first`, K+1, `terms, starting with k=0 are `): print(``): gu:=[seq(C2k(k1,t),k1=1..K)]: print(``): print(gu): print(``): print(` and in Maple format `): print(``): lprint(gu): print(``): print(`----------------------------------------------`): print(``): print(`This ends this article that took`, time(), `seconds to generate. `): print(``): end: #Paper3(K): outputs a paper with the recurrence for the Poincare series of Cbar(3,k) and the first K terms. Try: #Paper3(30); Paper3:=proc(K1) local gu,z,k,B3,k1,t,ope,INI,i,w,K: INI:= [1, 1/(t+1)/(t^2+t+1), (t^4-t^2+1)/(t-1)^4/(t+1)^4/(t^2+t+1)^4, (t^14-t^13-\ 2*t^12+6*t^11+6*t^10-9*t^9+t^8+17*t^7+t^6-9*t^5+6*t^4+6*t^3-2*t^2-t+1)/(t^2+t+1 )^7/(t+1)^8/(t-1)^10]: ope:= (3*k^8*t^28+90*k^8*t^27-82*k^7*t^28+685*k^8*t^26-2382*k^7*t^27+957*k^6*t^28-734 *k^8*t^25-19162*k^7*t^26+27250*k^6*t^27-6227*k^5*t^28-34686*k^8*t^24-1728*k^7*t ^25+231651*k^6*t^26-175497*k^5*t^27+24670*k^4*t^28-217754*k^8*t^23+757756*k^7*t ^24+272630*k^6*t^25-1574725*k^5*t^26+693391*k^4*t^27-60747*k^3*t^28-833782*k^8* t^22+5072024*k^7*t^23-6728055*k^6*t^24-3427475*k^5*t^25+6553590*k^4*t^26-\ 1712463*k^3*t^27+90290*k^2*t^28-2465960*k^8*t^21+19949404*k^7*t^22-49403333*k^6 *t^23+30358721*k^5*t^24+20198149*k^4*t^25-16997329*k^3*t^26+2562925*k^2*t^27-\ 73344*k*t^28-5939904*k^8*t^20+59970244*k^7*t^21-200861854*k^6*t^22+259097228*k^ 5*t^23-67832173*k^4*t^24-66099943*k^3*t^25+26606522*k^2*t^26-2100858*k*t^27+ 24480*t^28-12100988*k^8*t^19+146062188*k^7*t^20-615523168*k^6*t^21+1100596738*k ^5*t^22-781328432*k^4*t^23+39770751*k^3*t^24+122197523*k^2*t^25-22686384*k*t^26 +707544*t^27-21299821*k^8*t^18+299919432*k^7*t^19-1518197464*k^6*t^20+ 3453922572*k^5*t^21-3537005638*k^4*t^22+1325299718*k^3*t^23+123970858*k^2*t^24-\ 117631686*k*t^25+7885152*t^26-32799928*k^8*t^17+530789602*k^7*t^18-3145183561*k ^6*t^19+8647527252*k^5*t^20-11456026734*k^4*t^21+6670469386*k^3*t^22-1105218249 *k^2*t^23-237790932*k*t^24+44493264*t^25-44406725*k^8*t^16+820431334*k^7*t^17-\ 5600461437*k^6*t^18+18099947932*k^5*t^19-29224047838*k^4*t^20+22621365734*k^3*t ^21-6935724126*k^2*t^22+253147326*k*t^23+116229456*t^24-53206022*k^8*t^15+ 1113554282*k^7*t^16-8693210087*k^6*t^17+32458392739*k^5*t^18-61937044586*k^4*t^ 19+59180608644*k^3*t^20-25373023554*k^2*t^21+3403385376*k*t^22+82995696*t^23-\ 56496788*k^8*t^14+1336070020*k^7*t^15-11832368878*k^6*t^16+50630299438*k^5*t^17 -112017570300*k^4*t^18+127469585354*k^3*t^19-68906614906*k^2*t^20+14404725618*k *t^21-567591120*t^22-53206022*k^8*t^13+1419308600*k^7*t^14-14219189267*k^6*t^15 +69135688622*k^5*t^16-175760196548*k^4*t^17+233030410015*k^3*t^18-151814483737* k^2*t^19+41568336540*k*t^20-3251548800*t^21-44406725*k^8*t^12+1336070020*k^7*t^ 13-15112403600*k^6*t^14+83233657914*k^5*t^15-240919623835*k^4*t^16+368350036138 *k^3*t^17-281614417226*k^2*t^18+94743692490*k*t^19-10304835264*t^20-32799928*k^ 8*t^11+1113554282*k^7*t^12-14219189267*k^6*t^13+88512319328*k^5*t^14-\ 290682008584*k^4*t^15+507312002576*k^3*t^16-449579722789*k^2*t^17+179459858220* k*t^18-24613988688*t^19-21299821*k^8*t^10+820431334*k^7*t^11-11832368878*k^6*t^ 12+83233657914*k^5*t^13-309327096536*k^4*t^14+613774250822*k^3*t^15-\ 623070634106*k^2*t^16+290499818106*k*t^17-47907611712*t^18-12100988*k^8*t^9+ 530789602*k^7*t^10-8693210087*k^6*t^11+69135688622*k^5*t^12-290682008584*k^4*t^ 13+653697953696*k^3*t^14-756552417559*k^2*t^15+406070483856*k*t^16-78918640560* t^17-5939904*k^8*t^8+299919432*k^7*t^9-5600461437*k^6*t^10+50630299438*k^5*t^11 -240919623835*k^4*t^12+613774250822*k^3*t^13-806662390452*k^2*t^14+495505874844 *k*t^15-111487267008*t^16-2465960*k^8*t^7+146062188*k^7*t^8-3145183561*k^6*t^9+ 32458392739*k^5*t^10-175760196548*k^4*t^11+507312002576*k^3*t^12-756552417559*k ^2*t^13+529125151416*k*t^14-136866284856*t^15-833782*k^8*t^6+59970244*k^7*t^7-\ 1518197464*k^6*t^8+18099947932*k^5*t^9-112017570300*k^4*t^10+368350036138*k^3*t ^11-623070634106*k^2*t^12+495505874844*k*t^13-146419512768*t^14-217754*k^8*t^5+ 19949404*k^7*t^6-615523168*k^6*t^7+8647527252*k^5*t^8-61937044586*k^4*t^9+ 233030410015*k^3*t^10-449579722789*k^2*t^11+406070483856*k*t^12-136866284856*t^ 13-34686*k^8*t^4+5072024*k^7*t^5-200861854*k^6*t^6+3453922572*k^5*t^7-\ 29224047838*k^4*t^8+127469585354*k^3*t^9-281614417226*k^2*t^10+290499818106*k*t ^11-111487267008*t^12-734*k^8*t^3+757756*k^7*t^4-49403333*k^6*t^5+1100596738*k^ 5*t^6-11456026734*k^4*t^7+59180608644*k^3*t^8-151814483737*k^2*t^9+179459858220 *k*t^10-78918640560*t^11+685*k^8*t^2-1728*k^7*t^3-6728055*k^6*t^4+259097228*k^5 *t^5-3537005638*k^4*t^6+22621365734*k^3*t^7-68906614906*k^2*t^8+94743692490*k*t ^9-47907611712*t^10+90*k^8*t-19162*k^7*t^2+272630*k^6*t^3+30358721*k^5*t^4-\ 781328432*k^4*t^5+6670469386*k^3*t^6-25373023554*k^2*t^7+41568336540*k*t^8-\ 24613988688*t^9+3*k^8-2382*k^7*t+231651*k^6*t^2-3427475*k^5*t^3-67832173*k^4*t^ 4+1325299718*k^3*t^5-6935724126*k^2*t^6+14404725618*k*t^7-10304835264*t^8-82*k^ 7+27250*k^6*t-1574725*k^5*t^2+20198149*k^4*t^3+39770751*k^3*t^4-1105218249*k^2* t^5+3403385376*k*t^6-3251548800*t^7+957*k^6-175497*k^5*t+6553590*k^4*t^2-\ 66099943*k^3*t^3+123970858*k^2*t^4+253147326*k*t^5-567591120*t^6-6227*k^5+ 693391*k^4*t-16997329*k^3*t^2+122197523*k^2*t^3-237790932*k*t^4+82995696*t^5+ 24670*k^4-1712463*k^3*t+26606522*k^2*t^2-117631686*k*t^3+116229456*t^4-60747*k^ 3+2562925*k^2*t-22686384*k*t^2+44493264*t^3+90290*k^2-2100858*k*t+7885152*t^2-\ 73344*k+707544*t+24480)/(t+1)^4/(t-1)^6/(t^2+t+1)^3/(k-1)^2/(k^6*t^18+34*k^6*t^ 17-23*k^5*t^18+384*k^6*t^16-744*k^5*t^17+215*k^4*t^18+2010*k^6*t^15-8368*k^5*t^ 16+6701*k^4*t^17-1047*k^3*t^18+5818*k^6*t^14-44456*k^5*t^15+75140*k^4*t^16-\ 31746*k^3*t^17+2802*k^2*t^18+12520*k^6*t^13-131494*k^5*t^14+405560*k^4*t^15-\ 355308*k^3*t^16+83265*k^2*t^17-3904*k*t^18+24716*k^6*t^12-287134*k^5*t^13+ 1224616*k^4*t^14-1950531*k^3*t^15+931264*k^2*t^16-114342*k*t^17+2208*t^18+36422 *k^6*t^11-565932*k^5*t^12+2713321*k^4*t^13-6007266*k^3*t^14+5206549*k^2*t^15-\ 1279376*k*t^16+64008*t^17+47737*k^6*t^10-841120*k^5*t^11+5347239*k^4*t^12-\ 13502637*k^3*t^13+16340986*k^2*t^14-7295722*k*t^15+717120*t^16+51756*k^6*t^9-\ 1102583*k^5*t^10+8010408*k^4*t^11-26641068*k^3*t^12+37248184*k^2*t^13-23313476* k*t^14+4178688*t^15+47737*k^6*t^8-1195396*k^5*t^9+10505430*k^4*t^10-40204041*k^ 3*t^11+73656469*k^2*t^12-53859680*k*t^13+13580640*t^14+36422*k^6*t^7-1102583*k^ 5*t^8+11391892*k^4*t^9-52768275*k^3*t^10+111918787*k^2*t^11-106836144*k*t^12+ 31769016*t^13+24716*k^6*t^6-841120*k^5*t^7+10505430*k^4*t^8-57240594*k^3*t^9+ 147054359*k^2*t^10-163364954*k*t^11+63252792*t^12+12520*k^6*t^5-565932*k^5*t^6+ 8010408*k^4*t^7-52768275*k^3*t^8+159590222*k^2*t^9-214939672*k*t^10+97275120*t^ 11+5818*k^6*t^4-287134*k^5*t^5+5347239*k^4*t^6-40204041*k^3*t^7+147054359*k^2*t ^8-233382308*k*t^9+128182560*t^10+2010*k^6*t^3-131494*k^5*t^4+2713321*k^4*t^5-\ 26641068*k^3*t^6+111918787*k^2*t^7-214939672*k*t^8+139252176*t^9+384*k^6*t^2-\ 44456*k^5*t^3+1224616*k^4*t^4-13502637*k^3*t^5+73656469*k^2*t^6-163364954*k*t^7 +128182560*t^8+34*k^6*t-8368*k^5*t^2+405560*k^4*t^3-6007266*k^3*t^4+37248184*k^ 2*t^5-106836144*k*t^6+97275120*t^7+k^6-744*k^5*t+75140*k^4*t^2-1950531*k^3*t^3+ 16340986*k^2*t^4-53859680*k*t^5+63252792*t^6-23*k^5+6701*k^4*t-355308*k^3*t^2+ 5206549*k^2*t^3-23313476*k*t^4+31769016*t^5+215*k^4-31746*k^3*t+931264*k^2*t^2-\ 7295722*k*t^3+13580640*t^4-1047*k^3+83265*k^2*t-1279376*k*t^2+4178688*t^3+2802* k^2-114342*k*t+717120*t^2-3904*k+64008*t+2208)/K-(3*k^10*t^30+47*k^10*t^29-101* k^9*t^30-763*k^10*t^28-1547*k^9*t^29+1505*k^8*t^30-17000*k^10*t^27+22511*k^9*t^ 28+22822*k^8*t^29-13065*k^7*t^30-124197*k^10*t^26+527076*k^9*t^27-293255*k^8*t^ 28-198018*k^7*t^29+73110*k^6*t^30-550791*k^10*t^25+3979177*k^9*t^26-7266457*k^8 *t^27+2217612*k^7*t^28+1115235*k^6*t^29-275190*k^5*t^30-2033626*k^10*t^24+ 18140711*k^9*t^25-56741166*k^8*t^26+58608678*k^7*t^27-10758705*k^6*t^28-4245423 *k^5*t^29+704262*k^4*t^30-6406837*k^10*t^23+67756442*k^9*t^24-265750545*k^8*t^ 25+473806953*k^7*t^26-305943690*k^6*t^27+34906923*k^5*t^28+11022712*k^4*t^29-\ 1206700*k^3*t^30-16300174*k^10*t^22+214244865*k^9*t^23-1004745691*k^8*t^24+ 2278444488*k^7*t^25-2563081812*k^6*t^26+1078579164*k^5*t^27-76491017*k^4*t^28-\ 19195012*k^3*t^29+1319632*k^2*t^30-36127405*k^10*t^21+549041424*k^9*t^22-\ 3191255129*k^8*t^23+8724324921*k^7*t^24-12647765331*k^6*t^25+9373041192*k^5*t^ 26-2596127557*k^4*t^27+111409858*k^3*t^28+21347408*k^2*t^29-826944*k*t^30-\ 69426725*k^10*t^20+1221334153*k^9*t^21-8236611974*k^8*t^22+27855590115*k^7*t^23 -49068058065*k^6*t^24+47434687833*k^5*t^25-23426030577*k^4*t^26+4202775642*k^3* t^27-102834772*k^2*t^28-13598400*k*t^29+223488*t^30-117248564*k^10*t^19+ 2353331731*k^9*t^20-18392578626*k^8*t^21+72404195799*k^7*t^22-157599319752*k^6* t^23+186512960067*k^5*t^24-121510278441*k^4*t^25+39419174198*k^3*t^26-\ 4364634800*k^2*t^27+54105096*k*t^28+3730176*t^29-174835275*k^10*t^18+3982462208 *k^9*t^19-35540947501*k^8*t^20+162331021683*k^7*t^21-412535487093*k^6*t^22+ 602987051514*k^5*t^23-484334587058*k^4*t^24+209422828504*k^3*t^25-42599826792*k ^2*t^26+2612629440*k*t^27-12283488*t^28-232175099*k^10*t^17+5948235757*k^9*t^18 -60272964169*k^8*t^19+314629873575*k^7*t^20-928798010052*k^6*t^21+1589552896599 *k^5*t^22-1576970365862*k^4*t^23+846290888626*k^3*t^24-231619183260*k^2*t^25+ 26569994352*k*t^26-679130496*t^27-274526987*k^10*t^16+7905532763*k^9*t^17-\ 90175885438*k^8*t^18+534761516868*k^7*t^19-1805935200084*k^6*t^20+3594462729360 *k^5*t^21-4186599564827*k^4*t^22+2776366124346*k^3*t^23-948900778472*k^2*t^24+ 147692712720*k*t^25-7208585568*t^26-290361694*k^10*t^15+9353980963*k^9*t^16-\ 119957271763*k^8*t^17+801457100640*k^7*t^18-3076619403792*k^6*t^19+ 7012355871336*k^5*t^20-9510276903113*k^4*t^21+7423270719258*k^3*t^22-\ 3137738689132*k^2*t^23+613294555128*k*t^24-40906247040*t^25-274526987*k^10*t^14 +9895543862*k^9*t^15-142034365216*k^8*t^16+1067196717687*k^7*t^17-4619242985979 *k^6*t^18+11975502923838*k^5*t^19-18618059510662*k^4*t^20+16942113100508*k^3*t^ 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64008*t^17+47737*k^6*t^10-841120*k^5*t^11+5347239*k^4*t^12-13502637*k^3*t^13+ 16340986*k^2*t^14-7295722*k*t^15+717120*t^16+51756*k^6*t^9-1102583*k^5*t^10+ 8010408*k^4*t^11-26641068*k^3*t^12+37248184*k^2*t^13-23313476*k*t^14+4178688*t^ 15+47737*k^6*t^8-1195396*k^5*t^9+10505430*k^4*t^10-40204041*k^3*t^11+73656469*k ^2*t^12-53859680*k*t^13+13580640*t^14+36422*k^6*t^7-1102583*k^5*t^8+11391892*k^ 4*t^9-52768275*k^3*t^10+111918787*k^2*t^11-106836144*k*t^12+31769016*t^13+24716 *k^6*t^6-841120*k^5*t^7+10505430*k^4*t^8-57240594*k^3*t^9+147054359*k^2*t^10-\ 163364954*k*t^11+63252792*t^12+12520*k^6*t^5-565932*k^5*t^6+8010408*k^4*t^7-\ 52768275*k^3*t^8+159590222*k^2*t^9-214939672*k*t^10+97275120*t^11+5818*k^6*t^4-\ 287134*k^5*t^5+5347239*k^4*t^6-40204041*k^3*t^7+147054359*k^2*t^8-233382308*k*t ^9+128182560*t^10+2010*k^6*t^3-131494*k^5*t^4+2713321*k^4*t^5-26641068*k^3*t^6+ 111918787*k^2*t^7-214939672*k*t^8+139252176*t^9+384*k^6*t^2-44456*k^5*t^3+ 1224616*k^4*t^4-13502637*k^3*t^5+73656469*k^2*t^6-163364954*k*t^7+128182560*t^8 +34*k^6*t-8368*k^5*t^2+405560*k^4*t^3-6007266*k^3*t^4+37248184*k^2*t^5-\ 106836144*k*t^6+97275120*t^7+k^6-744*k^5*t+75140*k^4*t^2-1950531*k^3*t^3+ 16340986*k^2*t^4-53859680*k*t^5+63252792*t^6-23*k^5+6701*k^4*t-355308*k^3*t^2+ 5206549*k^2*t^3-23313476*k*t^4+31769016*t^5+215*k^4-31746*k^3*t+931264*k^2*t^2-\ 7295722*k*t^3+13580640*t^4-1047*k^3+83265*k^2*t-1279376*k*t^2+4178688*t^3+2802* k^2-114342*k*t+717120*t^2-3904*k+64008*t+2208)/K^3+4*t*(k-4)*(k-5)^2*(2*k-9)^2* (k^6*t^18+34*k^6*t^17-17*k^5*t^18+384*k^6*t^16-540*k^5*t^17+115*k^4*t^18+2010*k ^6*t^15-6064*k^5*t^16+3491*k^4*t^17-397*k^3*t^18+5818*k^6*t^14-32396*k^5*t^15+ 39060*k^4*t^16-11702*k^3*t^17+736*k^2*t^18+12520*k^6*t^13-96586*k^5*t^14+213430 *k^4*t^15-130748*k^3*t^16+21303*k^2*t^17-690*k*t^18+24716*k^6*t^12-212014*k^5*t ^13+654416*k^4*t^14-732651*k^3*t^15+238260*k^2*t^16-19762*k*t^17+252*t^18+36422 *k^6*t^11-417636*k^5*t^12+1465451*k^4*t^13-2307382*k^3*t^14+1373906*k^2*t^15-\ 221748*k*t^16+7176*t^17+47737*k^6*t^10-622588*k^5*t^11+2888319*k^4*t^12-5270293 *k^3*t^13+4439214*k^2*t^14-1322197*k*t^15+80856*t^16+51756*k^6*t^9-816161*k^5*t ^10+4351138*k^4*t^11-10417112*k^3*t^12+10336659*k^2*t^13-4377400*k*t^14+502098* t^15+47737*k^6*t^8-884860*k^5*t^9+5708570*k^4*t^10-15845169*k^3*t^11+20528119*k ^2*t^12-10378489*k*t^13+1699824*t^14+36422*k^6*t^7-816161*k^5*t^8+6191252*k^4*t ^9-20817645*k^3*t^10+31504242*k^2*t^11-20738818*k*t^12+4093590*t^13+24716*k^6*t ^6-622588*k^5*t^7+5708570*k^4*t^8-22591866*k^3*t^9+41472339*k^2*t^10-32084939*k *t^11+8238072*t^12+12520*k^6*t^5-417636*k^5*t^6+4351138*k^4*t^7-20817645*k^3*t^ 8+45042172*k^2*t^9-42340552*k*t^10+12830622*t^11+5818*k^6*t^4-212014*k^5*t^5+ 2888319*k^4*t^6-15845169*k^3*t^7+41472339*k^2*t^8-46022522*k*t^9+16979556*t^10+ 2010*k^6*t^3-96586*k^5*t^4+1465451*k^4*t^5-10417112*k^3*t^6+31504242*k^2*t^7-\ 42340552*k*t^8+18467748*t^9+384*k^6*t^2-32396*k^5*t^3+654416*k^4*t^4-5270293*k^ 3*t^5+20528119*k^2*t^6-32084939*k*t^7+16979556*t^8+34*k^6*t-6064*k^5*t^2+213430 *k^4*t^3-2307382*k^3*t^4+10336659*k^2*t^5-20738818*k*t^6+12830622*t^7+k^6-540*k ^5*t+39060*k^4*t^2-732651*k^3*t^3+4439214*k^2*t^4-10378489*k*t^5+8238072*t^6-17 *k^5+3491*k^4*t-130748*k^3*t^2+1373906*k^2*t^3-4377400*k*t^4+4093590*t^5+115*k^ 4-11702*k^3*t+238260*k^2*t^2-1322197*k*t^3+1699824*t^4-397*k^3+21303*k^2*t-\ 221748*k*t^2+502098*t^3+736*k^2-19762*k*t+80856*t^2-690*k+7176*t+252)/(t^2+t+1) ^3/(t+1)^6/(t-1)^14/(k-3)/(k-2)^2/(k-1)^2/(k^6*t^18+34*k^6*t^17-23*k^5*t^18+384 *k^6*t^16-744*k^5*t^17+215*k^4*t^18+2010*k^6*t^15-8368*k^5*t^16+6701*k^4*t^17-\ 1047*k^3*t^18+5818*k^6*t^14-44456*k^5*t^15+75140*k^4*t^16-31746*k^3*t^17+2802*k ^2*t^18+12520*k^6*t^13-131494*k^5*t^14+405560*k^4*t^15-355308*k^3*t^16+83265*k^ 2*t^17-3904*k*t^18+24716*k^6*t^12-287134*k^5*t^13+1224616*k^4*t^14-1950531*k^3* t^15+931264*k^2*t^16-114342*k*t^17+2208*t^18+36422*k^6*t^11-565932*k^5*t^12+ 2713321*k^4*t^13-6007266*k^3*t^14+5206549*k^2*t^15-1279376*k*t^16+64008*t^17+ 47737*k^6*t^10-841120*k^5*t^11+5347239*k^4*t^12-13502637*k^3*t^13+16340986*k^2* t^14-7295722*k*t^15+717120*t^16+51756*k^6*t^9-1102583*k^5*t^10+8010408*k^4*t^11 -26641068*k^3*t^12+37248184*k^2*t^13-23313476*k*t^14+4178688*t^15+47737*k^6*t^8 -1195396*k^5*t^9+10505430*k^4*t^10-40204041*k^3*t^11+73656469*k^2*t^12-53859680 *k*t^13+13580640*t^14+36422*k^6*t^7-1102583*k^5*t^8+11391892*k^4*t^9-52768275*k ^3*t^10+111918787*k^2*t^11-106836144*k*t^12+31769016*t^13+24716*k^6*t^6-841120* k^5*t^7+10505430*k^4*t^8-57240594*k^3*t^9+147054359*k^2*t^10-163364954*k*t^11+ 63252792*t^12+12520*k^6*t^5-565932*k^5*t^6+8010408*k^4*t^7-52768275*k^3*t^8+ 159590222*k^2*t^9-214939672*k*t^10+97275120*t^11+5818*k^6*t^4-287134*k^5*t^5+ 5347239*k^4*t^6-40204041*k^3*t^7+147054359*k^2*t^8-233382308*k*t^9+128182560*t^ 10+2010*k^6*t^3-131494*k^5*t^4+2713321*k^4*t^5-26641068*k^3*t^6+111918787*k^2*t ^7-214939672*k*t^8+139252176*t^9+384*k^6*t^2-44456*k^5*t^3+1224616*k^4*t^4-\ 13502637*k^3*t^5+73656469*k^2*t^6-163364954*k*t^7+128182560*t^8+34*k^6*t-8368*k ^5*t^2+405560*k^4*t^3-6007266*k^3*t^4+37248184*k^2*t^5-106836144*k*t^6+97275120 *t^7+k^6-744*k^5*t+75140*k^4*t^2-1950531*k^3*t^3+16340986*k^2*t^4-53859680*k*t^ 5+63252792*t^6-23*k^5+6701*k^4*t-355308*k^3*t^2+5206549*k^2*t^3-23313476*k*t^4+ 31769016*t^5+215*k^4-31746*k^3*t+931264*k^2*t^2-7295722*k*t^3+13580640*t^4-1047 *k^3+83265*k^2*t-1279376*k*t^2+4178688*t^3+2802*k^2-114342*k*t+717120*t^2-3904* k+64008*t+2208)/K^4: print(``): print(`A recurrence for the Poincare series of Allan Berele's Cbar(3,k) obtained via the Apagodu-Zeilberger multi-variable Almkvist-Zeilberger algorithm and using it to compute the first`, K1, `terms `): print(``): print(`By Shalosh B. Ekhad `): print(``): print(`Thanks to the Apagodu-Zeilberger multi-variable Almkvist-Zeilberger algorithm we have the following `): print(``): print(`Theorem: Let C3(k,t) be the Poincare series of Cbar(3,k) that according to Allan Berele equals (1/2)*1/(1-t)^(3*k) times B3(k,t), where B3(k,t) equals `): print(`the constant term, in z,w of `): print((1-z)*(1-1/z)*(1-w)*(1-1/w)*(1-z*w)*(1-1/(z*w))*((1-t/z)*(1-t*z)*(1-t/w)*(1-t*w)*(1-t*z*w)*(1-t/(z*w)))^(-k)): print(``): print(`One can compute many terms for B3(k,t) and hence C3(k,t) using the following fourth-order linear recurrence `): print(``): lprint(B3(k,t)=add(coeff(ope,K,-i)*B3(k-i,t),i=1..4)): print(``): print(`subject to the initial conditions`): print(``): lprint(seq(B3(i,t)=INI[i+1],i=0..3)): print(``): print(`using this recurrence, the first`, K1+1, `terms, starting with k=0 are `): print(``): gu:=[seq(C3k(k1,t),k1=1..K1)]: print(``): print(gu): print(``): print(` and in Maple format `): print(``): lprint(gu): print(``): print(`----------------------------------------------`): print(``): print(`This ends this article that took`, time(), `seconds to generate. `): print(``): end: