%I #11 Oct 05 2015 17:34:14

%S 1,21,661,28941,1678501,124467021,11484880261,1290503997741,

%T 173495416001701,27499205820027021,5075028072491665861,

%U 1078923766195953890541,261780612944688782844901,71901410584558939807059021,22195276604290979611365107461,7651037112318147566092161607341

%N a(n) = sum(stirling2(n,k)*(k+2)!*(k+3)!, k=1..n)/144.

%C It appears that for all n the last digit of a(n) is 1.

%F Representation as a sum of infinite series of special values of hypergeometric functions of type 2F0, in Maple notation: sum(k^n*(k+2)!*(k+3)!*hypergeom([k+3,k+4],[],-1)/k!, k=1..infinity)/144, n=1,2... .

%F a(n) ~ exp(1/2) * (n+2)! * (n+3)! / 144. - _Vaclav Kotesovec_, Oct 05 2015

%p with(combinat): a:= n-> sum(stirling2(n, k)*(k+2)!*(k+3)!, k=1..n)/144: seq(a(n), n=1..20);

%t Table[Sum[StirlingS2[n, k] (k + 2)! (k + 3)!, {k, n}]/144, {n, 16}] (* _Michael De Vlieger_, Oct 05 2015 *)

%Y Cf. A261833.

%K nonn

%O 1,2

%A _Karol A. Penson_ and Katarzyna Gorska, Oct 05 2015