A242229 a(n) = Sum_{k=0..n} k^(3*n) * k! * StirlingS2(n,k).

1, 1, 129, 121171, 421842405, 3921960731851, 80097035486409669, 3154805675402432477371, 218356776433458097793841045, 24765902586563160053438320367371, 4359137561016969073655241431827801509, 1139916274502953599866121961715757905518171

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..125

Formula

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...

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

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

Mathematica

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

Prog

(PARI) a(n) = sum(k=0, n, k!*k^(3*n)*stirling(n, k, 2)); \\ Seiichi Manyama, Feb 01 2022
(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
(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

Crossrefs

Cf. A203798, A242228

Cf. A000670, A122399, A229260.

Sequence in context: A176368 A283535 A183553 * A351137 A138586 A103347

Adjacent sequences: A242226 A242227 A242228 * A242230 A242231 A242232

Keyword

nonn,easy

Author

Vaclav Kotesovec, May 08 2014

Extensions

a(0)=1 prepended by Seiichi Manyama, Feb 01 2022

Status

approved