A242229 - OEIS

%I #22 Feb 01 2022 08:34:27

%S 1,1,129,121171,421842405,3921960731851,80097035486409669,

%T 3154805675402432477371,218356776433458097793841045,

%U 24765902586563160053438320367371,4359137561016969073655241431827801509,1139916274502953599866121961715757905518171

%N a(n) = Sum_{k=0..n} k^(3*n) * k! * StirlingS2(n,k).

%H Seiichi Manyama, <a href="/A242229/b242229.txt">Table of n, a(n) for n = 0..125</a>

%F a(n) ~ c * d^n * (n!)^4 / n^(3/2), where d = 20.5647332000203822461493845960846630764635... = r^4*(1+exp(3/r)), r = 0.97762267432285928683132021521727105447350... is the root of the equation (1+exp(-3/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.0600744446309702764688382302731840300640714536...

%F E.g.f.: Sum_{k>=0} (exp(k^3*x) - 1)^k. - _Seiichi Manyama_, Feb 01 2022

%F G.f.: Sum_{k>=0} k! * (k^3*x)^k/Product_{j=1..k} (1 - k^3*j*x). - _Seiichi Manyama_, Feb 01 2022

%t Table[Sum[k^(3*n) * k! * StirlingS2[n,k], {k,0,n}], {n,0,20}]

%o (PARI) a(n) = sum(k=0, n, k!*k^(3*n)*stirling(n, k, 2)); \\ _Seiichi Manyama_, Feb 01 2022

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (exp(k^3*x)-1)^k))) \\ _Seiichi Manyama_, Feb 01 2022

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k^3*x)^k/prod(j=1, k, 1-k^3*j*x))) \\ _Seiichi Manyama_, Feb 01 2022

%Y Cf. A203798, A242228

%Y Cf. A000670, A122399, A229260.

%K nonn,easy

%O 0,3

%A _Vaclav Kotesovec_, May 08 2014

%E a(0)=1 prepended by _Seiichi Manyama_, Feb 01 2022