%I #12 Nov 09 2024 08:09:00

%S 1,3,13,106,1273,20226,402589,9637902,269967793,8666441650,

%T 313793596981,12653878751526,562489374836041,27328756018660266,

%U 1440892788988703821,81940739770677315646,4999648556871348611425,325806859913842861709922,22584652022005415601772645

%N E.g.f. satisfies A(x) = (1 + x)^2 * exp(x * A(x)).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: (1+x)^2 * exp( -LambertW(-x*(1+x)^2) ).

%F E.g.f.: -LambertW(-x*(1+x)^2)/x.

%F a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(2*k+2,n-k)/k!.

%F a(n) ~ sqrt(1 + 3*r) * n^(n-1) / (exp(n - 1/4) * r^(n + 3/4)), where r = 0.2394629861788505554394435808448... is root of the equation r*(1+r)^2 = exp(-1). - _Vaclav Kotesovec_, Nov 09 2024

%o (PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(2*k+2, n-k)/k!);

%Y Cf. A377826, A377828.

%Y Cf. A362772, A377740, A377810.

%K nonn,new

%O 0,2

%A _Seiichi Manyama_, Nov 09 2024