CA for Concrete Mathematics, 2024
cschneid@risc.jku.at

Section 4

Symbolic summation

<<Sigma.m

mySum=SigmaSum[1/(k+1)SigmaBinomial[2k,k]/(n-k)SigmaBinomial[2n-2k-2,n-k-1],{k,0,A}]

sol=SigmaReduce[mySum]

sol=sol/.A->n-1//CollectProdSum

SigmaReduce[sol,n,Tower->{SigmaBinomial[2n,n]}]

GeneratingFunctions

Exit[]

<<RISC`GeneratingFunctions`

Package GeneratingFunctions version 0.8 written by Christian Mallinger
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

tree=GuessRE[{1,1,2,5,14,42},a[n]]

Guessing details

seqL={1,1,2,5,14,42};

recAnsatz=Sum[Sum[b[i,j]n^j ,{j,0,1}]a[n+i],{i,0,1}]

equL=Table[recAnsatz==0/.n->k/.a[A_]:>seqL[[A+1]],{k,0,4}]

sol=Solve[equL][[1]]

rec=recAnsatz/.sol

rec=(rec/.b[0,0]->2)==0

Continue...

de=RE2DE[tree[[1]],a[n],f[x]]

DSolve[de,f[x],x]

Section 5

Exit[]

<<RISC`GeneratingFunctions`

Package GeneratingFunctions version 0.8 written by Christian Mallinger
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

fSeries=Series[E^(-x),{x,0,10}]

fL=CoefficientList[fSeries,x]

rec=GuessRE[fL,f[n]]

RSolve[rec[[1]],f[n],n]