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 A378041 #10 Feb 16 2025 08:34:07
%S A378041 1,2,15,220,4873,145446,5479639,249736936,13366083889,821950542730,
%T A378041 57117681241471,4426656694204020,378577567656396409,
%U A378041 35416929943920575662,3598006167290727776263,394460149364865110384896,46420283015545052734709473,5836509710708683465245181458
%N A378041 E.g.f. satisfies A(x) = exp( x * A(x)^2 / (1-x) ) / (1-x).
%H A378041 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A378041 E.g.f.: exp( -LambertW(-2*x/(1-x)^3)/2 )/(1-x).
%F A378041 a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n+2*k,n-k)/k!.
%F A378041 a(n) ~ 3^(n + 5/3) * c^((n + 2)/3) * n^(n-1) / (exp(n) * (3*c^(1/3) - c^(2/3)*3^(1/3) * exp(1/3) + 2*3^(2/3) * exp(2/3))^n) / (sqrt(2) * (c^(2/3) - 2*3^(1/3) * exp(1/3))^(5/2) * sqrt((3^(2/3)*c^(2/3) - 6*exp(1/3)) / (9*3^(1/3)*c^(2/3) - 8*3^(1/3)*c^(2/3) * exp(1) + 8*3^(2/3)*exp(4/3) - 15*3^(1/6) * exp(1/3)*(c/sqrt(3)) + 2*c^(1/3)*exp(2/3) * (c + 15)))), where c = 9 + sqrt(81 + 24*exp(1)). - _Vaclav Kotesovec_, Nov 15 2024
%o A378041 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x/(1-x)^3)/2)/(1-x)))
%o A378041 (PARI) a(n) = n!*sum(k=0, n, (2*k+1)^(k-1)*binomial(n+2*k, n-k)/k!);
%Y A378041 Cf. A377595, A378042.
%K A378041 nonn
%O A378041 0,2
%A A378041 _Seiichi Manyama_, Nov 15 2024