A362690 - OEIS

%I #24 Nov 10 2023 07:02:49

%S 1,1,5,28,245,2816,40537,702976,14270153,332102656,8719631981,

%T 255020847104,8222803663549,289815184113664,11085650268060929,

%U 457386463819595776,20248713707077863953,957435459515190345728,48157934732749633188565

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

%C Essentially the same as A138293.

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

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

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

%F a(n) ~ sqrt(1 + LambertW(2*exp(-2))) * 2^((n+1)/2) * n^(n-1) / (exp(n) * LambertW(2*exp(-2))^((n+1)/2)). - _Vaclav Kotesovec_, Nov 10 2023

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x^2-lambertw(-x*exp(x^2)))))

%Y Cf. A349562, A362691.

%Y Cf. A125500, A138293, A362736.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 01 2023