# unsure of convergence to Integer sequence

Gerald McGarvey Gerald.McGarvey at comcast.net
Sun Dec 5 20:58:51 CET 2004

```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.

```