<< GeneratingFunctions.m

GeneratingFunctions Package by Christian Mallinger — © RISC Linz — V 0.69 (28-Sep-2009)

harmrec = {(n+2) h[n+2]-(2n+3)h[n+1]+(n+1)h[n]==0,h[0]==0,h[1]==1}

ode = RE2DE[harmrec,h[n],H[x]]

RE2DE[First[harmrec],h[n],H[x]]

DSolve[ode,H[x],x]

(* a[n] = Sum[k! HarmonicNumber[n-k],{k,0,n}] *)

fact = {a[n+1]-(n+1)a[n]==0,a[0]==1}

harmrec = harmrec/.h->a

proved = RECauchy[fact,harmrec,a[n]]

data = Table[Sum[k! HarmonicNumber[n-k],{k,0,n}],{n,0,40}];

GuessRE[data,a[n]]

?GuessRE

GuessRE[data,a[n],5,3]

data1 = -data;

guess = GuessRE[data1,a[n],5,3][[1]]

proved

REPlus[guess,proved,a[n]]

<< HolonomicFunctions.m

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

Annihilator[HarmonicNumber[n],{S[n]}]

ann=Annihilator[Sum[1/k,{k,1,n}],{S[n]}]

ApplyOreOperator[ann,h[n]]

Annihilator[Exp[x/Sqrt[1-4x]],{Der[x]}]//Factor

FullForm[%]

?DFinite*

?DFiniteSubstitute

DFiniteSubstitute[Annihilator[Exp[y],{Der[y]}],{y->x/Sqrt[1-4x]},Algebra->OreAlgebra[Der[x]]]

Annihilator[Binomial[n,k],{S[k],S[n]}]

Annihilator[StirlingS1[n,k],{S[k],S[n]}]

Annihilator[Sum[Binomial[m,k] StirlingS1[n,k],{k,0,m}],{S[m],S[n]}]

Annihilator[Sum[Binomial[n,k] StirlingS1[n,k],{k,0,n}],{S[n]}]

<<Guess.m

GuessMinRE[data1,a[n]]

dataS = Table[Sum[Binomial[m,k] StirlingS1[n,k],{k,0,m}],{m,0,20},{k,0,20}];

GuessMultRE[dataS,Flatten[Table[a[m+mm,k+kk],{mm,0,1},{kk,0,1}]],{m,k},1,Infolevel->3]

GuessMultRE[Table[Binomial[2n,k]^2,{k,0,10},{n,0,10}],Flatten[Table[a[k+kk,n+nn],{kk,0,1},{nn,0,1}]],{k,n},2,Infolevel->3]

ann=Annihilator[Binomial[2n,k]^2,{S[k],S[n]}]//Factor

FindRelation[ann,Support->{1,S[k],S[n]}]//Factor