A349883 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^3 * x).

1, 1, 5, 60, 1242, 41241, 2033683, 141318208, 13262986788, 1624337451945, 252725477615989, 48858277079478156, 11523986801592238046, 3265676705193282018577, 1097336766468309067029991, 432291795385094609190468384, 197690320046319097006619353352

Offset: 0

Author

Seiichi Manyama, Dec 03 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..227

Crossrefs

Cf. A031971, A349836, A349901.

Programs

Mathematica

a[n_] := Sum[If[k == 3*n - 2*k == 0, 1, k^(3*n - 2*k)], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Dec 04 2021 *)

PARI

a(n, t=3) = sum(k=0, n, k^(t*(n-k)+k));

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^3*x)))

Formula

a(n) = Sum_{k=0..n} k^(3*n-2*k).

a(n) ~ sqrt(Pi) * (3/2)^(1/2 + 3*n - 3*n/LambertW(3*exp(1)*n/2)) * (n/LambertW(3*exp(1)*n/2))^(1/2 + 3*n - 3*n/LambertW(3*exp(1)*n/2)) / sqrt(1 + LambertW(3*exp(1)*n/2)). - Vaclav Kotesovec, Dec 04 2021