######################################################################
##TSPP: Save this file as TSPP   To use it, stay in the              #
##same directory, get into Maple (by typing: maple <Enter> )         #
##and then type:  read TSPP : <Enter>                                #
##Then follow the instructions given there                           #
##                                                                   #
## Written by Manuel Kauers (RISC-Linz)                              #
## Christoph Koutschan (RISC-Linz)                                   #
## and Doron Zeilberger (Rutgers University)                         #
######################################################################
 
#Created: Aug. 2, 2008
 

print(`Created: Aug. 2, 2008.`):
print(`This version: Aug. 6, 2008. (adding procedure ManueljFree) `):
print(` This is  TSPP `):
print(`to semi-rigorously prove John Stembridge's Theorem`):
print(`about the enumeration of Totally Symmetric Plane Partitions`):
print(`whose 3D Ferrers diagram is bounded in an n by n by n box`):
print(` (originally conjectured by Ian Macdonald). `):
print(`It accompanies the paper `):
print(` "A Proof of George Andrews' and Dave Robbins' q-TSPP Conjecture`):
print(` (modulo a finite amount of routine calculations)" `):
print(`by Manuel Kauers, Christoph Koutschan, and Doron Zeilberger`):
print(` available from each of their websites.`):
print(`--------------------------------------`):
print(`Note: this program is mainly pedagogical and methodological reasons`):
print(`only, since there already exists a fully rigorous proof by John`):
print(`Stembridge`):
print(`But John Stembridge's human-generated proof is exteremely hard to`):
print(`follow (as are almost all non-trivial human-generated proofs`):
print(`and the present proof is conceptually must easier to follow`):
print(`since it reduces the statement to a finite amount `):
print(`of routine calculations.`):
print(`--------------------------------------`):
print(``):
print(`Please report bugs to zeilberg at math dot rutgers dot edu`):
print(``):
 print(`The most current version of this  package and paper`):
 print(` are  available from`):
 print(`http://www.math.rutgers.edu/~zeilberg/  .`):
 print(`For a list of the procedures type ezra();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):

with(combinat):

ezra1:=proc()

if args=NULL then
 print(` The supporting procedures are:`):
 print( ` `):
else
ezra(args):
fi:

end:



ezra:=proc()

if args=NULL then
 print(`The main procedure is: CheckTSPP`):
 print(``):
print(`The supporting procedurs are:`):
 print(` B, CheckManueljFree, CheckTSPPa, CheckTSPPb, ManueljFree, Okada`):
 print(`  `):



elif nops([args])=1 and op(1,[args])=B then
print(`B(n,j): inputs non-negative integers n and j `):
print(`and outputs the value`):
print(`of the holonomic function B(n,j) described in the article`):
print(` It is he promised`):
print(`full holonomic description of B(n,j) that we claim equals the `):
print(`normalized co-factor of the (n,j)-entry of the Okada matrix. `):
print(`For example, try: B(10,3); `);

elif nops([args])=1 and op(1,[args])=CheckManueljFree then
print(`CheckManueljFree(n0) :checks that the proposed j-free partial operator`):
print(`that is supposed to annihilate B(n,j) does indeed for all`):
print(`1<=j<=n<=n0. For example, try:`):
print(`CheckManueljFree(30);`):

elif nops([args])=1 and op(1,[args])=CheckTSPP then
print(`CheckTSPP(N0): does both CheckTSPPa and CheckTSPPb`):
print(`in other words:`):
print(`Given a pos. integer N0`):
print(`checks the validity of our claimed identities`):
print(`(Soichi) and (Okada) for all n <=N0. In the paper we prove that`):
print(`that there exists an N0 such that if CheckTSPP(N0) returns true`):
print(`then we have a fully (WZ-style) rigorous proof of Stembridge's `):
print(`Theorem. Due to compuational limitations, at present, we don't `):
print(`know its value, but it is very likely that N0=100 suffices.`):
print(`For example, try: `):
print(`CheckTSPP(20);`):

elif nops([args])=1 and op(1,[args])=CheckTSPPa then
print(`CheckTSPPa(N0): Given a pos. integer N0`):
print(`checks the validity of our claimed identity`):
print(`(Soichi) for all 0<= n <=n0. `):
print(`For example, try: `):
print(`CheckTSPPa(20);`):

elif nops([args])=1 and op(1,[args])=CheckTSPPb then
print(`CheckTSPPb(N0): Given a pos. integer N0`):
print(`checks the validity of our claimed identity`):
print(`(Okada) for all 0<=n <=n0. `):
print(`For example, try: `):
print(`CheckTSPPb(20);`):


elif nops([args])=1 and op(1,[args])=ManueljFree then
print(`ManueljFree: the j-free recurrence operator that annihilates B(n,j) evalaued`):
print(`at that point. For example, try:`):
print(`ManueljFree(10,5);`):


elif nops([args])=1 and op(1,[args])=Okada then
print(`Okada(i,j): The (i,j)-entry of the Okada matrix,`):
print(`whose n by n determinant equals the square of  the  `):
print(`number of Totally Symmetric Plane Partitions inside an`):
print(`n by n by n box. `):
print(`For example, try: Okada(4,3); `);

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




B:=proc(n,j) option remember:
if j<0 or n<0 or j>n then
  RETURN(0):
elif j=n then
  RETURN(1):
 elif j=n-1 then
    if n=2 then
      RETURN(-3/4):
    elif n=3 then
         RETURN(-32/25):
    elif n=4 then
         RETURN(-229/128):
   else
(-((-7 + 2*n)*(-8 + 3*n)*(-4 + 3*n)*(-29 + 193*n - 162*n^2 + 36*n^3)*
B(-3 + n, -4 + n)) + 
  (-5 + 2*n)*(-5 + 3*n)*(-1 + 3*n)*(-420 + 625*n - 270*n^2 + 36*n^3)*
B(n-2,n-3)
+ 
  (-3 + 2*n)*(-11 + 3*n)*(-7 + 3*n)*(-29 + 193*n - 162*n^2 + 36*n^3)*
B(-1 + n, -2 + n))/
 ((-1 + 2*n)*(-8 + 3*n)*(-4 + 3*n)*(-420 + 625*n - 270*n^2 + 36*n^3)):
fi:

else

((1 + j - n)*(1 + j + n)*(3040 + 5308*j + 3848*j^2 + 1325*j^3 + 222*j^4 + 15*j^5 - 928*n^2 - 589*j*n^2 - 
    134*j^2*n^2 - 11*j^3*n^2)*B(n, 1 + j) - 
(62000 + 158398*j + 172894*j^2 + 104130*j^3 + 37275*j^4 + 
    7925*j^5 + 931*j^6 + 47*j^7 - 23614*n^2 - 37572*j*n^2 - 
23789*j^2*n^2 - 7405*j^3*n^2 - 1148*j^4*n^2 - 
    72*j^5*n^2 + 1934*n^4 + 1322*j*n^4 + 317*j^2*n^4 + 27*j^3*n^4)*
B(n, 2 + j) + 
  (361056 + 749580*j + 670412*j^2 + 334314*j^3 + 100364*j^4 + 18152*j^5 + 1834*j^6 + 80*j^7 - 60888*n^2 - 
    84299*j*n^2 - 47370*j^2*n^2 - 13473*j^3*n^2 - 1942*j^4*n^2 - 114*j^5*n^2 + 2232*n^4 + 1703*j*n^4 + 
    446*j^2*n^4 + 41*j^3*n^4)*
B(n, 3 + j) - 
(927392 + 1662534*j + 1288776*j^2 + 560162*j^3 + 
    147366*j^4 + 23444*j^5 + 2086*j^6 + 80*j^7 - 
82570*n^2 - 105796*j*n^2 - 55635*j^2*n^2 - 14957*j^3*n^2 - 
    2048*j^4*n^2 - 114*j^5*n^2 + 1898*n^4 + 1486*j*n^4 + 
415*j^2*n^4 + 41*j^3*n^4)*B(n, 4 + j) + 
  (1165072 + 1865936*j + 1296280*j^2 + 506665*j^3 + 120050*j^4 + 
17186*j^5 + 1372*j^6 + 47*j^7 - 67400*n^2 - 
    82365*j*n^2 - 41164*j^2*n^2 - 10541*j^3*n^2 - 1372*j^4*n^2 - 
72*j^5*n^2 + 1048*n^4 + 853*j*n^4 + 
    250*j^2*n^4 + 27*j^3*n^4)*
B(n, 5 + j) - 
(6 + j - n)*(6 + j + n)*
   (19122 + 21702*j + 10159*j^2 + 2459*j^3 + 303*j^4 + 
15*j^5 - 402*n^2 - 330*j*n^2 - 97*j^2*n^2 - 
    11*j^3*n^2)*B(n, 6 + j) + 
2*(3 + j)*(14 + 6*j + j^2)*(6 + j - n)*(7 + j - n)*(6 + j + n)*(7 + j + n)*
   B(n, 7 + j))/
(2*(4 + j)*(21 + 8*j + j^2)*(j - n)*(1 + j - n)*(j + n)*(1 + j + n))  :

fi:

end:




with(LinearAlgebra):

An:=proc(n) local a,i,j:
a:=Matrix(n,n):

for i from 1 to n do
 for j from 1 to n do
  a[i,j]:=binomial(i+j-2,i-1)+binomial(i+j-1,i):
  if i=j then
   a[i,j]:=a[i,j]+ 2:
  elif i-j=1 then
      a[i,j]:=a[i,j]-1:
  fi:
 od:
od:

Determinant(a):

end:


Anj:=proc(n,J) local a,i,j:
a:=Matrix(n,n):

for i from 1 to n do
 for j from 1 to n do
  a[i,j]:=binomial(i+j-2,i-1)+binomial(i+j-1,i):
  if i=j then
   a[i,j]:=a[i,j]+ 2:
  elif i-j=1 then
      a[i,j]:=a[i,j]-1:
  fi:
 od:
od:

(-1)^(n+J)*Minor(a,n,J)/An(n-1):

end:

#CheckManuel(n0): checks Manuel's holonomic represetnation
#for the normalized co-factor of the (n,j) entry of the
#famous Okada determinant for 1<=j<=n<=n0
CheckManuel:=proc(n0) local n,j:

{seq(seq(evalb(Anj(n,j)=B(n,j)),j=1..n),n=1..n0)}:
end:


Okada:=proc(i,j) local gu:
gu:=binomial(i+j-2,i-1)+binomial(i+j-1,i):
  if i=j then
   gu:=gu+ 2:
  elif i-j=1 then
      gu:=gu-1:
  fi:
gu:
end:

CheckTSPPa:=proc(n0) local n,i,j:
evalb({seq(seq(add(B(n,j)*Okada(i,j),j=1..n),i=1..n-1),n=1..n0)}={0}):
end:


CheckTSPPb:=proc(n0) local gu,n,i,j:
gu:=[seq(add(B(n,j)*Okada(n,j),j=1..n),n=1..n0)]:
evalb({seq(-9/16*(3*i+5)^2*(3*i+1)^2/(2*i+3)^2/(2*i+1)^2*gu[i]+gu[i+2],
i=1..nops(gu)-2)}={0}):
end:

CheckTSPP:=proc(n0):
CheckTSPPa(n0) and CheckTSPPb(n0):
end:



#ManueljFree: the j-free recurrence operator that annihilates B(n,j) evalaued
#at that point. For example, try:
#ManueljFree(10,5);
ManueljFree:=proc(n,j) :
((1 + 3*n)^2*(4 + 3*n)*(5 + 3*n)^2*(7 + 3*n)*(8 + 3*n)*(11 + 3*n)*
   (337171903987518308569 + 312393901297681187688*n + 
    107500613689397571066*n^2 + 16287426506816589060*n^3 + 
    916634146678638660*n^4)*B(n, 4 + j))/741636239959645224960 - 
 ((1 + 3*n)^2*(4 + 3*n)*(5 + 3*n)^2*(7 + 3*n)*(8 + 3*n)*(11 + 3*n)*
   (337171903987518308569 + 312393901297681187688*n + 
    107500613689397571066*n^2 + 16287426506816589060*n^3 + 
    916634146678638660*n^4)*B(n, 5 + j))/185409059989911306240 + 
 ((1 + 3*n)^2*(4 + 3*n)*(5 + 3*n)^2*(7 + 3*n)*(8 + 3*n)*(11 + 3*n)*
   (337171903987518308569 + 312393901297681187688*n + 
    107500613689397571066*n^2 + 16287426506816589060*n^3 + 
    916634146678638660*n^4)*B(n, 6 + j))/123606039993274204160 - 
 ((1 + 3*n)^2*(4 + 3*n)*(5 + 3*n)^2*(7 + 3*n)*(8 + 3*n)*(11 + 3*n)*
   (337171903987518308569 + 312393901297681187688*n + 
    107500613689397571066*n^2 + 16287426506816589060*n^3 + 
    916634146678638660*n^4)*B(n, 7 + j))/185409059989911306240 + 
 ((1 + 3*n)^2*(4 + 3*n)*(5 + 3*n)^2*(7 + 3*n)*(8 + 3*n)*(11 + 3*n)*
   (337171903987518308569 + 312393901297681187688*n + 
    107500613689397571066*n^2 + 16287426506816589060*n^3 + 
    916634146678638660*n^4)*B(n, 8 + j))/741636239959645224960 + 
 ((7 + 2*n)*(4 + 3*n)^2*(7 + 3*n)*(8 + 3*n)^2*(10 + 3*n)*(11 + 3*n)*
   (14 + 3*n)*(210104368161284262893 + 145992645302545684324*n + 
    33191753857971350244*n^2 + 2469366266154925824*n^3)*B(1 + n, 3 + j))/
  1668681539909201756160 - ((7 + 2*n)*(4 + 3*n)^2*(7 + 3*n)*(8 + 3*n)^2*
   (10 + 3*n)*(11 + 3*n)*(14 + 3*n)*(210104368161284262893 + 
    145992645302545684324*n + 33191753857971350244*n^2 + 
    2469366266154925824*n^3)*B(1 + n, 4 + j))/370818119979822612480 + 
 ((7 + 2*n)*(4 + 3*n)^2*(7 + 3*n)*(8 + 3*n)^2*(10 + 3*n)*(11 + 3*n)*
   (14 + 3*n)*(210104368161284262893 + 145992645302545684324*n + 
    33191753857971350244*n^2 + 2469366266154925824*n^3)*B(1 + n, 5 + j))/
  278113589984866959360 - ((7 + 2*n)*(4 + 3*n)^2*(7 + 3*n)*(8 + 3*n)^2*
   (10 + 3*n)*(11 + 3*n)*(14 + 3*n)*(210104368161284262893 + 
    145992645302545684324*n + 33191753857971350244*n^2 + 
    2469366266154925824*n^3)*B(1 + n, 7 + j))/278113589984866959360 + 
 ((7 + 2*n)*(4 + 3*n)^2*(7 + 3*n)*(8 + 3*n)^2*(10 + 3*n)*(11 + 3*n)*
   (14 + 3*n)*(210104368161284262893 + 145992645302545684324*n + 
    33191753857971350244*n^2 + 2469366266154925824*n^3)*B(1 + n, 8 + j))/
  370818119979822612480 - ((7 + 2*n)*(4 + 3*n)^2*(7 + 3*n)*(8 + 3*n)^2*
   (10 + 3*n)*(11 + 3*n)*(14 + 3*n)*(210104368161284262893 + 
    145992645302545684324*n + 33191753857971350244*n^2 + 
    2469366266154925824*n^3)*B(1 + n, 9 + j))/1668681539909201756160 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 3*n)*(11 + 3*n)*(19944316464304087793196315 + 
    41618783878979564335942558*n + 37286995407636768064225517*n^2 + 
    18708141170407533801733812*n^3 + 5731477851989055830666373*n^4 + 
    1092497342867412363680094*n^5 + 125605894345926365731572*n^6 + 
    7873719869407186543176*n^7 + 202044359434512032928*n^8)*B(2 + n, 2 + j))/
  1251511154931901317120 - ((3 + 2*n)*(5 + 2*n)*(7 + 3*n)*(11 + 3*n)*
   (19944316464304087793196315 + 41618783878979564335942558*n + 
    37286995407636768064225517*n^2 + 18708141170407533801733812*n^3 + 
    5731477851989055830666373*n^4 + 1092497342867412363680094*n^5 + 
    125605894345926365731572*n^6 + 7873719869407186543176*n^7 + 
    202044359434512032928*n^8)*B(2 + n, 3 + j))/250302230986380263424 + 
 ((3 + 2*n)*(7 + 3*n)*(11 + 3*n)*(1051782692630755935943647645 + 
    2609734358684403520523950891*n + 2823108712511234586904611780*n^2 + 
    1743227160912602932451323427*n^3 + 674197519760177355136728318*n^4 + 
    168183775359713233170334071*n^5 + 26759158394080702963237299*n^6 + 
    2568385712501860357775646*n^7 + 129819730924404064779768*n^8 + 
    2378203814147890490640*n^9)*B(2 + n, 4 + j))/2503022309863802634240 + 
 ((3 + 2*n)*(7 + 3*n)*(11 + 3*n)*(444041042192050648546075980 + 
    1110003926168186438467630409*n + 1221979459430909947990578735*n^2 + 
    778493289097065144605477983*n^3 + 316907554251227846215264017*n^4 + 
    85697860915014369025664169*n^5 + 15436203967886145377033421*n^6 + 
    1790320108081469604909714*n^7 + 121545192115399933985112*n^8 + 
    3683126968887470497200*n^9)*B(2 + n, 5 + j))/625755577465950658560 - 
 ((3 + 2*n)*(7 + 3*n)*(11 + 3*n)*(2030174202826795018400098965 + 
    5065889644769101296265629167*n + 5553646191865730626922825112*n^2 + 
    3512282743962373869776274607*n^3 + 1413238363959005965943388474*n^4 + 
    375571679673249321435125019*n^5 + 65999781005909632023226599*n^6 + 
    7405023040516629463982310*n^7 + 481939207098108334712280*n^8 + 
    13878001938745579952592*n^9)*B(2 + n, 6 + j))/1251511154931901317120 + 
 ((3 + 2*n)*(7 + 3*n)*(11 + 3*n)*(444041042192050648546075980 + 
    1110003926168186438467630409*n + 1221979459430909947990578735*n^2 + 
    778493289097065144605477983*n^3 + 316907554251227846215264017*n^4 + 
    85697860915014369025664169*n^5 + 15436203967886145377033421*n^6 + 
    1790320108081469604909714*n^7 + 121545192115399933985112*n^8 + 
    3683126968887470497200*n^9)*B(2 + n, 7 + j))/625755577465950658560 + 
 ((3 + 2*n)*(7 + 3*n)*(11 + 3*n)*(1051782692630755935943647645 + 
    2609734358684403520523950891*n + 2823108712511234586904611780*n^2 + 
    1743227160912602932451323427*n^3 + 674197519760177355136728318*n^4 + 
    168183775359713233170334071*n^5 + 26759158394080702963237299*n^6 + 
    2568385712501860357775646*n^7 + 129819730924404064779768*n^8 + 
    2378203814147890490640*n^9)*B(2 + n, 8 + j))/2503022309863802634240 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 3*n)*(11 + 3*n)*(19944316464304087793196315 + 
    41618783878979564335942558*n + 37286995407636768064225517*n^2 + 
    18708141170407533801733812*n^3 + 5731477851989055830666373*n^4 + 
    1092497342867412363680094*n^5 + 125605894345926365731572*n^6 + 
    7873719869407186543176*n^7 + 202044359434512032928*n^8)*B(2 + n, 9 + j))/
  250302230986380263424 + ((3 + 2*n)*(5 + 2*n)*(7 + 3*n)*(11 + 3*n)*
   (19944316464304087793196315 + 41618783878979564335942558*n + 
    37286995407636768064225517*n^2 + 18708141170407533801733812*n^3 + 
    5731477851989055830666373*n^4 + 1092497342867412363680094*n^5 + 
    125605894345926365731572*n^6 + 7873719869407186543176*n^7 + 
    202044359434512032928*n^8)*B(2 + n, 10 + j))/1251511154931901317120 - 
 ((3 + 2*n)^2*(5 + 2*n)^2*(7 + 2*n)*(7 + 3*n)*(10 + 3*n)*(11 + 3*n)*
   (14 + 3*n)*(210104368161284262893 + 145992645302545684324*n + 
    33191753857971350244*n^2 + 2469366266154925824*n^3)*B(3 + n, 1 + j))/
  938633366198925987840 + (11*(3 + 2*n)^2*(5 + 2*n)^2*(7 + 2*n)*(7 + 3*n)*
   (10 + 3*n)*(11 + 3*n)*(14 + 3*n)*(210104368161284262893 + 
    145992645302545684324*n + 33191753857971350244*n^2 + 
    2469366266154925824*n^3)*B(3 + n, 2 + j))/1877266732397851975680 - 
 ((3 + 2*n)*(5 + 2*n)*(10 + 3*n)*(14 + 3*n)*(21378412698952678460183400 + 
    48281333922743618608614690*n + 47946230303341119020795102*n^2 + 
    27347288975891819307642582*n^3 + 9787075364556775720483005*n^4 + 
    2245863927929254671364758*n^5 + 321908064446090816256204*n^6 + 
    26281773361292188839048*n^7 + 933420448606561961472*n^8)*B(3 + n, 3 + j))/
  417170384977300439040 + ((3 + 2*n)*(5 + 2*n)*(10 + 3*n)*(14 + 3*n)*
   (390360823047484159221062250 + 790014434629762668960756310*n + 
    692548103383631641950302204*n^2 + 344392309386772795986387014*n^3 + 
    106513727299633942021725535*n^4 + 21025571130750485182828146*n^5 + 
    2591843761605660641137188*n^6 + 182704549022471335619736*n^7 + 
    5644971284430160433664*n^8)*B(3 + n, 4 + j))/2503022309863802634240 - 
 ((3 + 2*n)*(5 + 2*n)*(10 + 3*n)*(14 + 3*n)*(117663186360878842460094780 + 
    229099372790966554088796082*n + 190716026807438642542689698*n^2 + 
    88531803598103843640798158*n^3 + 24988755263651380511820805*n^4 + 
    4367673382027535664674694*n^5 + 457302988639300787193708*n^6 + 
    25774627558662594337608*n^7 + 577831706280252642816*n^8)*B(3 + n, 5 + j))/
  625755577465950658560 + ((3 + 2*n)*(5 + 2*n)*(10 + 3*n)*(14 + 3*n)*
   (117663186360878842460094780 + 229099372790966554088796082*n + 
    190716026807438642542689698*n^2 + 88531803598103843640798158*n^3 + 
    24988755263651380511820805*n^4 + 4367673382027535664674694*n^5 + 
    457302988639300787193708*n^6 + 25774627558662594337608*n^7 + 
    577831706280252642816*n^8)*B(3 + n, 7 + j))/625755577465950658560 - 
 ((3 + 2*n)*(5 + 2*n)*(10 + 3*n)*(14 + 3*n)*(390360823047484159221062250 + 
    790014434629762668960756310*n + 692548103383631641950302204*n^2 + 
    344392309386772795986387014*n^3 + 106513727299633942021725535*n^4 + 
    21025571130750485182828146*n^5 + 2591843761605660641137188*n^6 + 
    182704549022471335619736*n^7 + 5644971284430160433664*n^8)*
   B(3 + n, 8 + j))/2503022309863802634240 + 
 ((3 + 2*n)*(5 + 2*n)*(10 + 3*n)*(14 + 3*n)*(21378412698952678460183400 + 
    48281333922743618608614690*n + 47946230303341119020795102*n^2 + 
    27347288975891819307642582*n^3 + 9787075364556775720483005*n^4 + 
    2245863927929254671364758*n^5 + 321908064446090816256204*n^6 + 
    26281773361292188839048*n^7 + 933420448606561961472*n^8)*B(3 + n, 9 + j))/
  417170384977300439040 - (11*(3 + 2*n)^2*(5 + 2*n)^2*(7 + 2*n)*(7 + 3*n)*
   (10 + 3*n)*(11 + 3*n)*(14 + 3*n)*(210104368161284262893 + 
    145992645302545684324*n + 33191753857971350244*n^2 + 
    2469366266154925824*n^3)*B(3 + n, 10 + j))/1877266732397851975680 + 
 ((3 + 2*n)^2*(5 + 2*n)^2*(7 + 2*n)*(7 + 3*n)*(10 + 3*n)*(11 + 3*n)*
   (14 + 3*n)*(210104368161284262893 + 145992645302545684324*n + 
    33191753857971350244*n^2 + 2469366266154925824*n^3)*B(3 + n, 11 + j))/
  938633366198925987840 - ((3 + 2*n)*(5 + 2*n)^2*(7 + 2*n)^2*(9 + 2*n)*
   (11 + 2*n)*(13 + 3*n)*(17 + 3*n)*(59250724219255993109 + 
    57964187349094921864*n + 18236345509426743024*n^2 + 
    1858276835035833384*n^3)*B(4 + n, j))/703975024649194490880 + 
 ((3 + 2*n)*(5 + 2*n)^2*(7 + 2*n)^2*(9 + 2*n)*(11 + 2*n)*(13 + 3*n)*
   (17 + 3*n)*(59250724219255993109 + 57964187349094921864*n + 
    18236345509426743024*n^2 + 1858276835035833384*n^3)*B(4 + n, 1 + j))/
  117329170774865748480 - ((3 + 2*n)*(5 + 2*n)^2*(7 + 2*n)^2*(13 + 3*n)*
   (17 + 3*n)*(19247687231678442694149 + 26654068948040964421998*n + 
    14313980929470823103250*n^2 + 3749185038551302477201*n^3 + 
    480721210130049719946*n^4 + 24195111667983668088*n^5)*B(4 + n, 2 + j))/
  234658341549731496960 - ((3 + 2*n)*(5 + 2*n)^2*(7 + 2*n)^2*(13 + 3*n)*
   (17 + 3*n)*(6780976979734448413254 + 9231111835514113611862*n + 
    5028711770602283160342*n^2 + 1350524625208058616149*n^3 + 
    177877379440394222178*n^4 + 9178845737625664632*n^5)*B(4 + n, 3 + j))/
  140795004929838898176 + ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*
   (8985295286011072437300557148 + 21179431454074825693702574972*n + 
    22106080582634838372784754532*n^2 + 13410228911742404829834908380*n^3 + 
    5208967768782517832082770313*n^4 + 1342990659882683472696714774*n^5 + 
    229739732691434414816320962*n^6 + 25137843250933312695060444*n^7 + 
    1596191872766221037614968*n^8 + 44811346655912609258928*n^9)*
   B(4 + n, 4 + j))/3754533464795703951360 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(2894513495286480328685791668 + 
    6147097933151790357207520220*n + 5782875715194462449177951148*n^2 + 
    3177291411251036447159914036*n^3 + 1128197331637881245427066009*n^4 + 
    269490669633860435324732094*n^5 + 43447660153757730813826386*n^6 + 
    4570960251464602736213628*n^7 + 285226684888776260263992*n^8 + 
    8045724740157571546800*n^9)*B(4 + n, 5 + j))/938633366198925987840 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(5042626147549013914793022204 + 
    9449570354315027746013072580*n + 7584204220713587309983026420*n^2 + 
    3412893377535996967496859572*n^3 + 946045531250232803057679771*n^4 + 
    167525700412410635877539938*n^5 + 19220456534296967248634502*n^6 + 
    1456889420597402553314580*n^7 + 75228069185844861169512*n^8 + 
    2270267182405547579664*n^9)*B(4 + n, 6 + j))/1877266732397851975680 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(2894513495286480328685791668 + 
    6147097933151790357207520220*n + 5782875715194462449177951148*n^2 + 
    3177291411251036447159914036*n^3 + 1128197331637881245427066009*n^4 + 
    269490669633860435324732094*n^5 + 43447660153757730813826386*n^6 + 
    4570960251464602736213628*n^7 + 285226684888776260263992*n^8 + 
    8045724740157571546800*n^9)*B(4 + n, 7 + j))/938633366198925987840 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(8985295286011072437300557148 + 
    21179431454074825693702574972*n + 22106080582634838372784754532*n^2 + 
    13410228911742404829834908380*n^3 + 5208967768782517832082770313*n^4 + 
    1342990659882683472696714774*n^5 + 229739732691434414816320962*n^6 + 
    25137843250933312695060444*n^7 + 1596191872766221037614968*n^8 + 
    44811346655912609258928*n^9)*B(4 + n, 8 + j))/3754533464795703951360 - 
 ((3 + 2*n)*(5 + 2*n)^2*(7 + 2*n)^2*(13 + 3*n)*(17 + 3*n)*
   (6780976979734448413254 + 9231111835514113611862*n + 
    5028711770602283160342*n^2 + 1350524625208058616149*n^3 + 
    177877379440394222178*n^4 + 9178845737625664632*n^5)*B(4 + n, 9 + j))/
  140795004929838898176 - ((3 + 2*n)*(5 + 2*n)^2*(7 + 2*n)^2*(13 + 3*n)*
   (17 + 3*n)*(19247687231678442694149 + 26654068948040964421998*n + 
    14313980929470823103250*n^2 + 3749185038551302477201*n^3 + 
    480721210130049719946*n^4 + 24195111667983668088*n^5)*B(4 + n, 10 + j))/
  234658341549731496960 + ((3 + 2*n)*(5 + 2*n)^2*(7 + 2*n)^2*(9 + 2*n)*
   (11 + 2*n)*(13 + 3*n)*(17 + 3*n)*(59250724219255993109 + 
    57964187349094921864*n + 18236345509426743024*n^2 + 
    1858276835035833384*n^3)*B(4 + n, 11 + j))/117329170774865748480 - 
 ((3 + 2*n)*(5 + 2*n)^2*(7 + 2*n)^2*(9 + 2*n)*(11 + 2*n)*(13 + 3*n)*
   (17 + 3*n)*(59250724219255993109 + 57964187349094921864*n + 
    18236345509426743024*n^2 + 1858276835035833384*n^3)*B(4 + n, 12 + j))/
  703975024649194490880 + ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)^2*
   (11 + 2*n)*(16 + 3*n)*(20 + 3*n)*(84184227291135535391 + 
    80036479166255007976*n + 24561476197268387892*n^2 + 
    2469366266154925824*n^3)*B(5 + n, 1 + j))/703975024649194490880 - 
 (11*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)^2*(11 + 2*n)*(16 + 3*n)*
   (20 + 3*n)*(84184227291135535391 + 80036479166255007976*n + 
    24561476197268387892*n^2 + 2469366266154925824*n^3)*B(5 + n, 2 + j))/
  1407950049298388981760 + ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*
   (84971973893500012473165557 + 169184815813879532493706606*n + 
    145363379179141450810985294*n^2 + 70529673569021734030469550*n^3 + 
    21170350224930495930175479*n^4 + 4031322078197041501591062*n^5 + 
    476233294281479987520564*n^6 + 31952345095244348673768*n^7 + 
    933420448606561961472*n^8)*B(5 + n, 3 + j))/312877788732975329280 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(240686928275456540661572439 + 
    506086760263506869303474722*n + 464236149320137778147317088*n^2 + 
    243360847872628546065551150*n^3 + 79916422598142402857727133*n^4 + 
    16857596850118624788448434*n^5 + 2231751558656419241520348*n^6 + 
    169473214976582962672056*n^7 + 5644971284430160433664*n^8)*
   B(5 + n, 4 + j))/1877266732397851975680 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(-56674781698117406187069027 - 
    102097964359580410767282002*n - 75959622794399500200846694*n^2 - 
    29609073275187702828624730*n^3 - 6140061365617931531225009*n^4 - 
    513341000707224787484538*n^5 + 35480693755903719071124*n^6 + 
    10275064819193357456040*n^7 + 577831706280252642816*n^8)*B(5 + n, 5 + j))/
  469316683099462993920 - ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*
   (-56674781698117406187069027 - 102097964359580410767282002*n - 
    75959622794399500200846694*n^2 - 29609073275187702828624730*n^3 - 
    6140061365617931531225009*n^4 - 513341000707224787484538*n^5 + 
    35480693755903719071124*n^6 + 10275064819193357456040*n^7 + 
    577831706280252642816*n^8)*B(5 + n, 7 + j))/469316683099462993920 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(240686928275456540661572439 + 
    506086760263506869303474722*n + 464236149320137778147317088*n^2 + 
    243360847872628546065551150*n^3 + 79916422598142402857727133*n^4 + 
    16857596850118624788448434*n^5 + 2231751558656419241520348*n^6 + 
    169473214976582962672056*n^7 + 5644971284430160433664*n^8)*
   B(5 + n, 8 + j))/1877266732397851975680 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(84971973893500012473165557 + 
    169184815813879532493706606*n + 145363379179141450810985294*n^2 + 
    70529673569021734030469550*n^3 + 21170350224930495930175479*n^4 + 
    4031322078197041501591062*n^5 + 476233294281479987520564*n^6 + 
    31952345095244348673768*n^7 + 933420448606561961472*n^8)*B(5 + n, 9 + j))/
  312877788732975329280 + (11*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)^2*
   (11 + 2*n)*(16 + 3*n)*(20 + 3*n)*(84184227291135535391 + 
    80036479166255007976*n + 24561476197268387892*n^2 + 
    2469366266154925824*n^3)*B(5 + n, 10 + j))/1407950049298388981760 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)^2*(11 + 2*n)*(16 + 3*n)*
   (20 + 3*n)*(84184227291135535391 + 80036479166255007976*n + 
    24561476197268387892*n^2 + 2469366266154925824*n^3)*B(5 + n, 11 + j))/
  703975024649194490880 + ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*
   (11 + 2*n)^2*(-86581211911030062641604 - 98398859664584370741844*n - 
    36512718258381407464019*n^2 - 2647076074339310249322*n^3 + 
    1419991405965458995572*n^4 + 339674034777145156872*n^5 + 
    22449373270501336992*n^6)*B(6 + n, 2 + j))/703975024649194490880 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)^2*
   (-86581211911030062641604 - 98398859664584370741844*n - 
    36512718258381407464019*n^2 - 2647076074339310249322*n^3 + 
    1419991405965458995572*n^4 + 339674034777145156872*n^5 + 
    22449373270501336992*n^6)*B(6 + n, 3 + j))/140795004929838898176 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)*
   (-22237454627006828222255844 - 32334560164243026552726040*n - 
    18727302200506843778908083*n^2 - 5445959931704323838021321*n^3 - 
    793665938886305034676359*n^4 - 40634603631832310684478*n^5 + 
    2413303639778060810856*n^6 + 264244868238654498960*n^7)*B(6 + n, 4 + j))/
  1407950049298388981760 + ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*
   (11 + 2*n)*(7951554661686867886391184 + 13501311962255703501073660*n + 
    9750737897936380425089628*n^2 + 3913810831686895422962621*n^3 + 
    948552238640426461466079*n^4 + 139280561549025031435518*n^5 + 
    11481063993169014498984*n^6 + 409236329876385610800*n^7)*B(6 + n, 5 + j))/
  351987512324597245440 - ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)*
   (17187910667911288835770044 + 31715086725839693079116536*n + 
    25063150352609591710153605*n^2 + 11026429581719219675193803*n^3 + 
    2917936989139869383566773*n^4 + 463876465008431697323706*n^5 + 
    40927230208215678641544*n^6 + 1542000215416175550288*n^7)*
   B(6 + n, 6 + j))/703975024649194490880 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)*
   (7951554661686867886391184 + 13501311962255703501073660*n + 
    9750737897936380425089628*n^2 + 3913810831686895422962621*n^3 + 
    948552238640426461466079*n^4 + 139280561549025031435518*n^5 + 
    11481063993169014498984*n^6 + 409236329876385610800*n^7)*B(6 + n, 7 + j))/
  351987512324597245440 + ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)*
   (-22237454627006828222255844 - 32334560164243026552726040*n - 
    18727302200506843778908083*n^2 - 5445959931704323838021321*n^3 - 
    793665938886305034676359*n^4 - 40634603631832310684478*n^5 + 
    2413303639778060810856*n^6 + 264244868238654498960*n^7)*B(6 + n, 8 + j))/
  1407950049298388981760 - ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*
   (11 + 2*n)^2*(-86581211911030062641604 - 98398859664584370741844*n - 
    36512718258381407464019*n^2 - 2647076074339310249322*n^3 + 
    1419991405965458995572*n^4 + 339674034777145156872*n^5 + 
    22449373270501336992*n^6)*B(6 + n, 9 + j))/140795004929838898176 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)^2*
   (-86581211911030062641604 - 98398859664584370741844*n - 
    36512718258381407464019*n^2 - 2647076074339310249322*n^3 + 
    1419991405965458995572*n^4 + 339674034777145156872*n^5 + 
    22449373270501336992*n^6)*B(6 + n, 10 + j))/703975024649194490880 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)*(11 + 2*n)*(13 + 2*n)*(13 + 3*n)*
   (17 + 3*n)*(84184227291135535391 + 80036479166255007976*n + 
    24561476197268387892*n^2 + 2469366266154925824*n^3)*B(7 + n, 3 + j))/
  703975024649194490880 + ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)*
   (11 + 2*n)*(13 + 2*n)*(13 + 3*n)*(17 + 3*n)*(84184227291135535391 + 
    80036479166255007976*n + 24561476197268387892*n^2 + 
    2469366266154925824*n^3)*B(7 + n, 4 + j))/156438894366487664640 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)*(11 + 2*n)*(13 + 2*n)*(13 + 3*n)*
   (17 + 3*n)*(84184227291135535391 + 80036479166255007976*n + 
    24561476197268387892*n^2 + 2469366266154925824*n^3)*B(7 + n, 5 + j))/
  117329170774865748480 + ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)*
   (11 + 2*n)*(13 + 2*n)*(13 + 3*n)*(17 + 3*n)*(84184227291135535391 + 
    80036479166255007976*n + 24561476197268387892*n^2 + 
    2469366266154925824*n^3)*B(7 + n, 7 + j))/117329170774865748480 - 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)*(11 + 2*n)*(13 + 2*n)*(13 + 3*n)*
   (17 + 3*n)*(84184227291135535391 + 80036479166255007976*n + 
    24561476197268387892*n^2 + 2469366266154925824*n^3)*B(7 + n, 8 + j))/
  156438894366487664640 + ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)^2*(9 + 2*n)*
   (11 + 2*n)*(13 + 2*n)*(13 + 3*n)*(17 + 3*n)*(84184227291135535391 + 
    80036479166255007976*n + 24561476197268387892*n^2 + 
    2469366266154925824*n^3)*B(7 + n, 9 + j))/703975024649194490880 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)*(13 + 2*n)*(15 + 2*n)*
   (16 + 3*n)*(20 + 3*n)*(5635307114262861196 + 5292672435537820029*n + 
    1587534807380050233*n^2 + 152772357779773110*n^3)*B(8 + n, 4 + j))/
  175993756162298622720 - ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)*
   (13 + 2*n)*(15 + 2*n)*(16 + 3*n)*(20 + 3*n)*(5635307114262861196 + 
    5292672435537820029*n + 1587534807380050233*n^2 + 152772357779773110*n^3)*
   B(8 + n, 5 + j))/43998439040574655680 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)*(13 + 2*n)*(15 + 2*n)*
   (16 + 3*n)*(20 + 3*n)*(5635307114262861196 + 5292672435537820029*n + 
    1587534807380050233*n^2 + 152772357779773110*n^3)*B(8 + n, 6 + j))/
  29332292693716437120 - ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)*
   (13 + 2*n)*(15 + 2*n)*(16 + 3*n)*(20 + 3*n)*(5635307114262861196 + 
    5292672435537820029*n + 1587534807380050233*n^2 + 152772357779773110*n^3)*
   B(8 + n, 7 + j))/43998439040574655680 + 
 ((3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11 + 2*n)*(13 + 2*n)*(15 + 2*n)*
   (16 + 3*n)*(20 + 3*n)*(5635307114262861196 + 5292672435537820029*n + 
    1587534807380050233*n^2 + 152772357779773110*n^3)*B(8 + n, 8 + j))/
  175993756162298622720
:

end:



#CheckManueljFree(n0) :checks that the proposed j-free partial operator
#that is supposed to annihilate B(n,j) does indeed for all
#1<=j<=n<=n0. For example, try:
#CheckManueljFree(30);
CheckManueljFree:=proc(n0)  local n,j:
evalb({seq(seq(ManueljFree(n,j),j=1..n),n=1..n0)}={0}):
end: