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 A376578 #14 Feb 16 2025 08:34:07
%S A376578 1,1,3,16,149,1656,22567,367144,7020729,153727408,3792553451,
%T A376578 104109493104,3149158146037,104087591605336,3732676321011471,
%U A376578 144352230854996056,5988802115089340273,265333292818927985376,12503715930615717645139,624511093178389790592352
%N A376578 E.g.f. A(x) satisfies A(x) = exp(x*(1 + x^3)*A(x)).
%H A376578 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A376578 a(n) = n! * Sum_{k=0..floor(n/4)} (n-3*k+1)^(n-3*k-1) * binomial(n-3*k,k)/(n-3*k)!.
%F A376578 E.g.f.: exp( -LambertW(-x*(1+x^3)) ).
%F A376578 From _Vaclav Kotesovec_, Oct 10 2024: (Start)
%F A376578 E.g.f.: -LambertW(-x*(1+x^3))/(x*(1+x^3)).
%F A376578 a(n) ~ sqrt(1 + 4*r^3) * n^(n-1) / (exp(n - 3/2) * r^(n - 1/2)), where r = 0.3524487947753533587882657130849651130602232209406... is the positive real root of the equation r*(1 + r^3) = exp(-1). (End)
%o A376578 (PARI) a(n) = n!*sum(k=0, n\4, (n-3*k+1)^(n-3*k-1)*binomial(n-3*k, k)/(n-3*k)!);
%o A376578 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x^3)))))
%Y A376578 Cf. A362771, A376577.
%K A376578 nonn
%O A376578 0,3
%A A376578 _Seiichi Manyama_, Sep 28 2024