#File inDone 
read EKHAD:
print(`This is file outDone, proving Identity (Done) of the paper`):
print(`It was obtained by running file inDone`):
print(`Proof of the Refined Alternating Sign Matrix Conjecture by`):
print(`Doron Zeilberger, available as file refined.tex (or refined.ps)`):
print(`from htpp://www.math.temple.edu/~zeilberg`): 
 
 
Fb:=w^(-r-n)*binomial(n+r,n)*binomial(2*n-r,n)*(1+w*X)^r*(1-w^2*X)^(n-r)*
(w-1/w)^n:
 
G1:=((1-X^3)/(1-X))^(2*n+1)*rf(k+1,n)*rf(k+2/3,n)*X^(3*k)*(-1)^n/3^(n+1)
/n!^3/rf(-n+1/3,2*n+1)*(3*n+1)!:
 
G2:=((1-X^3)/(1-X))^(2*n+1)*rf(k+1,n)*rf(k+4/3,n)*X^(3*k+1)*(-1)^n/3^(n+1)
/n!^3/rf(-n+1/3,2*n+1)*(3*n+1)!:
 
print(`The identity (Done) is trivial when n=0`):
print(`Let's first prove it when n=1`):
print(`The difference of the left and right sides of (Done) when n=1`):
print(`is:`):
G11:=subs(n=1,G1):
G11:=simplify(G11):
sumG1:=sum(G11,k=0..infinity):
 
G21:=subs(n=1,G2):
G21:=simplify(G21):
sumG2:=sum(G21,k=0..infinity):
 
F1:=subs(n=1,Fb):
sumF1:=sum(F1,r=0..1):
sumF1:=subs(w=exp(Pi*I/3),sumF1):
sumF1:=expand(evalc(expand(sumF1))):
print(normal(sumF1-sumG1+sumG2)):
print(`This completes the proof when n=0,1`):
 
print(`Now let Shalosh find recurrences`):
 
print(`The summand of the right side of (Done) is`):
 
print(Fb):
 
ope1:=zeil(Fb,r,n)[1]:
 
print(`Using the program zeil we get the following operator, where w`):
print(`is arbitrary`):
print(ope1):
print(`The proof, using zeilpap(F,r,n) will be reproduced below`):
print(`Substituting w=exp(Pi*I/3), we get the operator`):
ope1:=subs(w=exp(Pi*I/3),ope1):
ope1:=evalc(ope1):
ope1:=expand(ope1):
ope1:=evalc(ope1):
ope1:=expand(ope1):
ku:=coeff(ope1,N,0):
ku1:=normal(coeff(ope1,N,1)/ku):
ku2:=normal(coeff(ope1,N,2)/ku):
ope1:=1+factor(ku1)*N+factor(ku2)*N^2:
print(ope1):
 
print(`The summand of the first sum on the left is`):
 
print(G1):
print(`Using zeil(G1,k,n) we get the following operator annihilating`):
print(`the first sum on the left`):
 
 
ope2:=zeil(G1,k,n)[1]:
ku:=coeff(ope2,N,0):
ku1:=normal(coeff(ope2,N,1)/ku):
ku2:=normal(coeff(ope2,N,2)/ku):
ope2:=1+factor(ku1)*N+factor(ku2)*N^2:
 
 print(ope2):
 
print(`The verbose proof will follow soon`):
 
print(`The summand of the second sum on the left is`):
 
print(G2):
print(`Using zeil(G2,k,n) we get the following operator annihilating`):
print(`the second sum`):
 
zeil(G2,k,n):
ope3:="[1]:
ku:=coeff(ope3,N,0):
ku1:=normal(coeff(ope3,N,1)/ku):
ku2:=normal(coeff(ope3,N,2)/ku):
ope3:=1+factor(ku1)*N+factor(ku2)*N^2:
print(ope3):
print(`To sum up, the three operators  are`):
print(ope1):
print(ope2):
print(ope3):
 
 
print(`While the outputs of zeil is completely proved, here are the`):
print(`the verbose proofs`):
 
zeilpap(Fb,r,n,`Right of (Done)`,`Paper refined.tex Doron Z.'s collection`):
zeilpap(G1,k,n,`Left1 of (Done)`,`Paper refined.tex Doron Z.'s collection`):
zeilpap(G2,k,n,`Left2 of (Done)`,`Paper refined.tex Doron Z.'s collection`):
 
print(`The whole thing took`,time(),`seconds of CPU time`):
 
quit: