A349878 Expansion of Sum_{k>=0} k^3 * x^k/(1 - k * x).

0, 1, 9, 44, 178, 689, 2723, 11304, 49772, 232657, 1151781, 6018628, 33087022, 190780001, 1150653679, 7241710656, 47454745496, 323154695841, 2282779990113, 16700904488284, 126356632389834, 987303454928465, 7957133905608283, 66071772829246808

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

Formula

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

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

Mathematica

Table[Sum[k^(n - k + 3), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2021 *)

Prog

(PARI) a(n, s=3, t=1) = sum(k=0, n, k^(t*(n-k)+s));
(PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^3*x^k/(1-k*x))))

Crossrefs

Cf. A026898, A003101, A062809, A349879.

Cf. A349854.

Sequence in context: A059825 A074631 A084903 * A034558 A144109 A099867

Adjacent sequences: A349875 A349876 A349877 * A349879 A349880 A349881

Keyword

nonn

Author

Seiichi Manyama, Dec 03 2021

Status

approved