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 A375541 #16 Jun 16 2025 19:19:04
%S A375541 1,2,20,318,7080,202650,7089516,293122998,13984363728,756140833458,
%T A375541 45695657262420,3052241497352718,223293580147036152,
%U A375541 17756185491727424586,1524930579202933587132,140665424413881644688870,13870450317973173165542304,1455954744856343617306729314
%N A375541 a(n) = 2^n * n! * [x^n] (-1/2 - exp(-x))^n.
%F A375541 a(n) ~ n^n / (sqrt(1+LambertW(exp(-1)/2)) * exp(n) * (LambertW(exp(-1)/2))^n). - _Vaclav Kotesovec_, Sep 01 2024
%F A375541 a(n) = Sum_{k=0..n} k^n*2^k*binomial(n,k). - _Ridouane Oudra_, Jun 16 2025
%p A375541 gf := n -> (-1/2 - exp(-x))^n:
%p A375541 ser := n -> series(gf(n), x, 20):
%p A375541 a := n -> expand(2^n*n!*coeff(ser(n), x, n)):
%p A375541 seq(a(n), n = 0..17);
%t A375541 Table[2^n * n! * SeriesCoefficient[(-1/2 - E^(-x))^n,{x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 01 2024 *)
%K A375541 nonn
%O A375541 0,2
%A A375541 _Peter Luschny_, Sep 01 2024