the Sum(k=1..Inf; k^n/CatalanNumber[k]) can be written in closed form (W. Gosper, 2000) as
Sum[Hypergeometric2F1[m+1,m+2,m+1/2,1/4]StirlingS2[n,m]/(2m-1)!!/2^m(m+1)!m!,{m,1,n}]
and this simplifies to (for n= 0..14)
2+(4*Pi)/(9*Sqrt[3])
2+(16*Pi)/(27*Sqrt[3])
(2*(567+52*Sqrt[3]*Pi))/243
14+(104*Pi)/(27*Sqrt[3])
158/3+(392*Pi)/(27*Sqrt[3])
238+(15944*Pi)/(243*Sqrt[3])
3758/3+(83912*Pi)/(243*Sqrt[3])
7518+(1510808*Pi)/(729*Sqrt[3])
151934/3+(30532552*Pi)/(2187*Sqrt[3])
378254+(228041096*Pi)/(2187*Sqrt[3])
9304142/3+(5609264024*Pi)/(6561*Sqrt[3])
27689662+(150241428616*Pi)/(19683*Sqrt[3])
802115870/3+(483578128856*Pi)/(6561*Sqrt[3])
2776117230+(15062943707464*Pi)/(19683*Sqrt[3])
92521462766/3+(167337504140872*Pi)/(19683*Sqrt[3])
Now, these expression *seem* to huddle uncomfortably close to integers:
2,806133
3,074844
6,995495
20,986486
79,000346
357,009124
1879,002190
11276,988463
75966,991041
567381,021008
4652071,037121
41534492,955918
401057934,821915
4164175845,053300
46260731383,985200
but, the loss of accuracy towards the end troubles me.
Can anyone suggest a mathematical basis for this small 'numirical' ?
W.
