A358687 a(n) = n! * Sum_{k=0..n} k^(3 * (n-k)) / (n-k)!.

1, 1, 4, 57, 1444, 61785, 4050126, 373648513, 47101090744, 7764843893265, 1630744323319450, 426925697290933401, 136591846585403311620, 52602030074554601172649, 24058544668572618782040022, 12916480280574798627072144465

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..221

Formula

E.g.f.: Sum_{k>=0} x^k * exp(k^3 * x).

G.f.: Sum_{k>=0} k! * x^k / (1 - k^3 * x)^(k+1).

log(a(n)) ~ (6*n*(log(n) - 1) + 3*log(1 + LambertW(n^(2/3))) + 2*n*LambertW(n^(2/3)) * (7*log(n) - 6*log(1 + LambertW(n^(2/3))) + 3*LambertW(n^(2/3)))) / (6*(1 + LambertW(n^(2/3)))). - Vaclav Kotesovec, Nov 27 2022

Mathematica

Join[{1}, Table[n! * Sum[k^(3*(n-k))/(n-k)!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 27 2022 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, k^(3*(n-k))/(n-k)!);
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k*exp(x)^k^3)))
(PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k/(1-k^3*x)^(k+1)))

Crossrefs

Cf. A006153, A193421, A358688.

Cf. A349880, A356673.

Sequence in context: A299562 A299459 A300103 * A209317 A231491 A240887

Adjacent sequences: A358684 A358685 A358686 * A358688 A358689 A358690

Keyword

nonn

Author

Seiichi Manyama, Nov 26 2022

Status

approved