A351134 - OEIS

%I #17 Feb 04 2022 08:27:25

%S 1,1,127,115028,383611414,3407421330934,66396378581670602,

%T 2493320561997330821496,164454446238949941359354760,

%U 17769323863754938530919641304080,2978930835291629440372517431365668448,741834782450714229554166000654848368247568

%N a(n) = Sum_{k=0..n} k! * k^(3*n) * Stirling1(n,k).

%H Seiichi Manyama, <a href="/A351134/b351134.txt">Table of n, a(n) for n = 0..125</a>

%F E.g.f.: Sum_{k>=0} log(1 + k^3*x)^k.

%F a(n) ~ c * d^n * n^(4*n + 1/2), where d = 0.358437102792682941192966771107499325675345706113923587904567864366079667... and c = 2.68150179193269103258189978938660205530269361522513... - _Vaclav Kotesovec_, Feb 04 2022

%t a[0] = 1; a[n_] := Sum[k! * k^(3*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 12, 0] (* _Amiram Eldar_, Feb 02 2022 *)

%o (PARI) a(n) = sum(k=0, n, k!*k^(3*n)*stirling(n, k, 1));

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k^3*x)^k)))

%Y Cf. A006252, A320083, A351133.

%Y Cf. A242229, A351135, A351137.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 02 2022