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 A372154 #17 Aug 05 2025 06:47:24
%S A372154 1,2,12,98,1128,16442,293356,6195114,151432112,4209004466,
%T A372154 131188519764,4533821784098,172125130420744,7122734349079338,
%U A372154 319148172778019708,15395906192167996058,795673541794111734624,43862837291529529270370
%N A372154 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x) * A(x)^(1/2) ).
%H A372154 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A372154 E.g.f.: A(x) = exp( -2 * LambertW(-x * (1+x)) ).
%F A372154 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 A372154 a(n) ~ sqrt(2 + 8*exp(-1) - 2*sqrt(1 + 4*exp(-1))) * 2^n * n^(n-1) / ((sqrt(1+4*exp(-1)) - 1)^n * exp(n - 5/2)). - _Vaclav Kotesovec_, Aug 05 2025
%o A372154 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(-x*(1+x)))))
%o A372154 (PARI) a(n, r=2, s=1, t=1, u=0) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(s*k, n-k)/k!);
%Y A372154 Cf. A362771, A372155.
%K A372154 nonn
%O A372154 0,2
%A A372154 _Seiichi Manyama_, Apr 20 2024