A365283 - OEIS

%I #12 Nov 08 2023 05:06:44

%S 1,1,2,12,120,1380,19440,341040,7029120,164762640,4355769600,

%T 128527439040,4181332700160,148633442717760,5734427199621120,

%U 238676208285715200,10659325532663808000,508452777299622355200,25800664274991135129600

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

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

%F a(n) ~ 2^(n/2) * (1 + 3*LambertW(2^(1/3)/3))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(2^(1/3)/3)) * 3^(3*n/2 + 2) * exp(n) * LambertW(2^(1/3)/3)^(3*(n+1)/2)). - _Vaclav Kotesovec_, Nov 08 2023

%t Join[{1},Table[n!/(n+1) * Sum[(n-2*k)^k * Binomial[n+1,n-2*k]/k!, {k, 0, Floor[n/2]}], {n,1,20}]] (* _Vaclav Kotesovec_, Nov 08 2023 *)

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

%Y Cf. A358064, A365282, A365284.

%Y Cf. A161633, A365287.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 31 2023