Something to note, the terms can be rewritten as follows: 2/1+(4*Pi)/(3^2*Sqrt[3]) 2/1+(16*Pi)/(3^3*Sqrt[3]) 14/3+(104*Pi)/(3^4*Sqrt[3]) 14/1+(936*Pi)/(3^5*Sqrt[3]) 158/3+(10584*Pi)/(3^6*Sqrt[3]) 238/1+(143496*Pi)/(3^7*Sqrt[3]) 3758/3+(2265624*Pi)/(3^8*Sqrt[3]) 7518/1+(40791816*Pi)/(3^9*Sqrt[3]) 151934/3+(824378904*Pi)/(3^10*Sqrt[3]) 378254/1+(18471328776*Pi)/(3^11*Sqrt[3]) 9304142/3+(454350385944*Pi)/(3^12*Sqrt[3]) 27689662/1+(12169555717896*Pi)/(3^13*Sqrt[3]) 802115870/3+(352528455936024*Pi)/(3^14*Sqrt[3]) 2776117230/1+(10980885962741256*Pi)/(3^15*Sqrt[3]) 92521462766/3+(365967121556087064*Pi)/(3^16*Sqrt[3]) which contains the following sequence: 4,16,104,936,10584,143496,2265624,40791816,824378904,18471328776,454350385944,12169555717896,352528455936024,10980885962741256,365967121556087064 Gerald At 10:41 AM 12/5/2004, wouter meeussen wrote:

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.