Takeuchi-Prellberg Constant

Author

Eric W. Weisstein
August 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/Takeuchi-PrellbergConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Takeuchi-PrellbergConstant.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Takeuchi Numbers

t[0]=0;
t[1]=1;
t[2]=4;
t[3]=14;
t[n_]:=t[n]=Sum[(2Binomial[n+k-1,k]-Binomial[n+k,k])t[n-k-1],{k,0,n-2}]+Sum[CatalanNumber[k],{k,n}]

t/@Range[0,20]

Sum[CatalanNumber[k],{k,n}]//FunctionExpand

FullSimplify[Sum[CatalanNumber[k],{k,n}],n∈Integers&&n>0]

Table[-(1/2)-((I (Sqrt[3]))/2)-Binomial[2 (1+n),1+n] Gamma[2+n] Hypergeometric2F1Regularized[(3/2)+n,1,3+n,4],{n,0,20}]//FunctionExpand//Simplify

Constant

Table[((t[n])/(BellB[n]Exp[((LambertW[n])^2)/2])),{n,20}]//N

SequenceLimit[N[Table[((t[n])/(BellB[n]Exp[((LambertW[n])^2)/2])),{n,1000}],200]]

ListPlot[Table[((t[n])/(BellB[n]Exp[((LambertW[n])^2)/2])),{n,20}],Joined->True]