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A349878
Expansion of Sum_{k>=0} k^3 * x^k/(1 - k * x).
3
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
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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Dec 03 2021
STATUS
approved