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 A360596 #17 Feb 16 2025 08:34:04
%S A360596 1,3,22,282,5224,126120,3742704,131612432,5347866752,246490091136,
%T A360596 12704900911360,724072211436288,45209213973292032,3068872654856532992,
%U A360596 225023336997933996032,17724257054969009940480,1492513932494133333753856,133800772458366199028023296
%N A360596 Expansion of e.g.f. 1/( (1 - x) * (1 + LambertW(-2*x)) ).
%H A360596 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A360596 a(n) = n! * Sum_{k=0..n} (2*k)^k / k!.
%F A360596 a(0)=1; a(n) = n*a(n-1) + (2*n)^n.
%F A360596 a(n) ~ 2^(n+1) * n^n / (2 - exp(-1)). - _Vaclav Kotesovec_, Feb 13 2023
%o A360596 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/((1-x)*(1+lambertw(-2*x)))))
%o A360596 (PARI) a(n) = n!*sum(k=0, n, (2*k)^k/k!);
%o A360596 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+(2*i)^i); v;
%Y A360596 Cf. A062971, A277506, A277509.
%K A360596 nonn,easy
%O A360596 0,2
%A A360596 _Seiichi Manyama_, Feb 13 2023