<<zb.m

Fast Zeilberger Package by Peter Paule and Markus Schorn (enhanced by Axel Riese) — © RISC Linz — V 3.54 (02/23/05)

T[n_]:=Sum[Binomial[n-j,n-2j],{j,0,Floor[n/2]}]

Table[T[n],{n,0,10}]

(* Mathematica knows some simplifications for T[n] already (by table look-up: *)

T[n]

(* but also Zeilberger knows what it is (note the natural boundaries!) and discovers with proof *)

Zb[Binomial[n-j,n-2j],{j,0,Infinity},n]

Prove[]

Quit

<< HolonomicFunctions.m

HolonomicFunctions package by Christoph Koutschan, RISC-Linz, Version 1.5.1 (09.08.2011)
⟶ Type  ?HolonomicFunctions  for help

ann=Annihilator[Binomial[n-j,n-2j],{S[j],S[n]}]

Factor[ann]

ApplyOreOperator[ann,f[n,j]]

{P,Q}=CreativeTelescoping[ann,S[j]-1,{S[n]}]

Factor[Q]

Annihilator[Sum[Binomial[n+k,2k](-4)^(-k),{k,0,n}],{S[n]}]

ann2F1=Annihilator[Hypergeometric2F1[-n,n+1,1,(1-x)/2],{S[n],Der[x]}]

FindRelation[ann2F1,Support->{1,S[n],S[n]^2}]

ann2F1=Annihilator[Hypergeometric2F1[-n,n+1,1,(1-x)/2],{Der[x],S[n]}]

annInt=Annihilator[Integrate[Hypergeometric2F1[-n,n+1,1,(1-x)/2],{x,-1,y}],{S[n],Der[y]}]

FindRelation[annInt,Support->{1,S[n],S[n]^2}]

(* rooks *)

f[x_,y_]:=(1/(1-(x/(1-x))-(y/(1-y))));

f[x,y]//Factor

g[x_,y_]:=(1/(1-2 x-2 y+3 x y))

annXY=Annihilator[g[x,y],{Der[x],Der[y]}]

annIJ = DFiniteDE2RE[annXY,{x,y},{i,j}]//Factor

First[Solve[ApplyOreOperator[annIJ,b[i,j]][[1]]==0,b[i+1,j+1]]]/.{i->i-1,j->j-1}

ClearAll[a];
a[0,0]:=1;
a[i_Integer,j_Integer]:=a[i,j]=Sum[a[i,k1],{k1,0,j-1}]+Sum[a[k2,j],{k2,0,i-1}]

<<Guess.m

data = Table[a[nn,nn],{nn,0,30}];

GuessMinRE[data,α[n]]