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 A372156 #17 Aug 05 2025 07:41:56
%S A372156 1,2,8,44,328,3032,33964,445580,6727984,114892784,2192201044,
%T A372156 46233324788,1068561369352,26865052934696,730137962157244,
%U A372156 21334636036296668,667074635111434336,22225983296836137440,786215841115748129956,29429693502599243538884
%N A372156 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x * A(x)^(1/2)) ).
%H A372156 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A372156 E.g.f.: A(x) = exp( 2*x - 2*LambertW(-x^2 * exp(x)) ).
%F A372156 If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(s*k,n-k)/k!.
%F A372156 a(n) ~ sqrt(1 + LambertW(exp(-1/2)/2)) * n^(n-1) / (LambertW(exp(-1/2)/2)^(n+4) * 2^(n + 5/2) * exp(n)). - _Vaclav Kotesovec_, Aug 05 2025
%o A372156 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*x-2*lambertw(-x^2*exp(x)))))
%o A372156 (PARI) a(n, r=2, s=1, t=0, u=1) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(s*k, n-k)/k!);
%Y A372156 Cf. A125500, A372157.
%K A372156 nonn
%O A372156 0,2
%A A372156 _Seiichi Manyama_, Apr 20 2024