A062809 -id:A062809 - cp's OEIS frontend

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

Seiichi Manyama, Dec 03 2021

nonn

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

Cf. A026898, A003101, A062809, A349879.

Cf. A349854.

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

a(n, s=3, t=1) = sum(k=0, n, k^(t*(n-k)+s));

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^3*x^k/(1-k*x))))

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

A349879 Expansion of Sum_{k>=0} k^4 * x^k/(1 - k * x).

Original entry on oeis.org

0, 1, 17, 114, 564, 2507, 10961, 49260, 231928, 1150781, 6017297, 33085294, 190777804, 1150650935, 7241707281, 47454741400, 323154690928, 2282779984281, 16700904481425, 126356632381834, 987303454919204, 7957133905597635, 66071772829234641

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

In general, for s>=1, Sum_{k=0..n} k^(n-k+s) ~ sqrt(2*Pi) * ((n + s)/LambertW(exp(1)*(n + s)))^(1/2 + (n + s)*(1 - 1/LambertW(exp(1)*(n + s)))) / sqrt(1 + LambertW(exp(1)*(n + s))). - Vaclav Kotesovec, Dec 04 2021

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

Cf. A026898, A003101, A062809, A349878.

Cf. A349855.

Table[Sum[k^(n - k + 4), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, s=4, t=1) = sum(k=0, n, k^(t*(n-k)+s));

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^4*x^k/(1-k*x))))

a(n) = Sum_{k=0..n} k^(n-k+4).

a(n) ~ sqrt(2*Pi) * ((n + 4)/LambertW(exp(1)*(n + 4)))^(1/2 + (n + 4)*(1 - 1/LambertW(exp(1)*(n + 4)))) / sqrt(1 + LambertW(exp(1)*(n + 4))). - Vaclav Kotesovec, Dec 04 2021

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

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A349879 Expansion of Sum_{k>=0} k^4 * x^k/(1 - k * x).

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Mathematica

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