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

%I A349879 #17 Mar 24 2025 10:59:33
%S A349879 0,1,17,114,564,2507,10961,49260,231928,1150781,6017297,33085294,
%T A349879 190777804,1150650935,7241707281,47454741400,323154690928,
%U A349879 2282779984281,16700904481425,126356632381834,987303454919204,7957133905597635,66071772829234641
%N A349879 Expansion of Sum_{k>=0} k^4 * x^k/(1 - k * x).
%C A349879 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
%H A349879 Seiichi Manyama, <a href="/A349879/b349879.txt">Table of n, a(n) for n = 0..500</a>
%F A349879 a(n) = Sum_{k=0..n} k^(n-k+4).
%F A349879 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
%t A349879 Table[Sum[k^(n - k + 4), {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 04 2021 *)
%o A349879 (PARI) a(n, s=4, t=1) = sum(k=0, n, k^(t*(n-k)+s));
%o A349879 (PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^4*x^k/(1-k*x))))
%Y A349879 Cf. A026898, A003101, A062809, A349878.
%Y A349879 Cf. A349855.
%K A349879 nonn
%O A349879 0,3
%A A349879 _Seiichi Manyama_, Dec 03 2021