This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349878 #17 Dec 07 2021 09:15:42
%S A349878 0,1,9,44,178,689,2723,11304,49772,232657,1151781,6018628,33087022,
%T A349878 190780001,1150653679,7241710656,47454745496,323154695841,
%U A349878 2282779990113,16700904488284,126356632389834,987303454928465,7957133905608283,66071772829246808
%N A349878 Expansion of Sum_{k>=0} k^3 * x^k/(1 - k * x).
%H A349878 Seiichi Manyama, <a href="/A349878/b349878.txt">Table of n, a(n) for n = 0..500</a>
%F A349878 a(n) = Sum_{k=0..n} k^(n-k+3).
%F A349878 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
%t A349878 Table[Sum[k^(n - k + 3), {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 04 2021 *)
%o A349878 (PARI) a(n, s=3, t=1) = sum(k=0, n, k^(t*(n-k)+s));
%o A349878 (PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^3*x^k/(1-k*x))))
%Y A349878 Cf. A026898, A003101, A062809, A349879.
%Y A349878 Cf. A349854.
%K A349878 nonn
%O A349878 0,3
%A A349878 _Seiichi Manyama_, Dec 03 2021