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 A365053 #18 Feb 16 2025 08:34:06
%S A365053 1,1,4,25,230,2786,42112,764296,16209916,393678856,10777609556,
%T A365053 328466815964,11031378197776,404830360798072,16118917055902312,
%U A365053 692126238230304616,31882272572881781648,1568365865590875789824,82061348851406564851312
%N A365053 E.g.f. satisfies A(x) = exp( x * (1+x/2) * A(x) ).
%H A365053 Seiichi Manyama, <a href="/A365053/b365053.txt">Table of n, a(n) for n = 0..379</a>
%H A365053 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A365053 E.g.f.: exp( -LambertW(-x * (1+x/2)) ).
%F A365053 a(n) = n! * Sum_{k=0..n} (1/2)^(n-k) * (k+1)^(k-1) * binomial(k,n-k)/k!.
%F A365053 From _Vaclav Kotesovec_, Nov 10 2023: (Start)
%F A365053 E.g.f.: -LambertW(-x * (1+x/2)) / (x * (1+x/2)).
%F A365053 a(n) ~ sqrt(-sqrt(1 + 2*exp(-1)) + 1 + 2*exp(-1)) * n^(n-1) / (exp(n - 3/2) * (-1 + sqrt(1 + 2*exp(-1)))^n). (End)
%o A365053 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x/2)))))
%Y A365053 Cf. A365054, A365055, A365056.
%Y A365053 Cf. A091485, A143740.
%K A365053 nonn,easy
%O A365053 0,3
%A A365053 _Seiichi Manyama_, Aug 19 2023