A364981 - OEIS

%I #12 Nov 18 2023 05:09:10

%S 1,1,4,39,580,11685,298566,9248701,336886936,14112113049,668422303210,

%T 35325208755441,2060811941835780,131547166492534117,

%U 9120279070776381886,682489450793082237285,54828316394224735284016,4706545644403274325580593

%N E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^3).

%F a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*n-2*k+1,k)/( (3*n-2*k+1)*(n-k)! ).

%F a(n) ~ sqrt((1 + r*s^3)/(12*s + 9*r*s^4)) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.1811100305436879929789759231994897963241226689807... and s = 1.522012903517407628213363540403002787906223513104... are real roots of the system of equations 1 + exp(r*s^3)*r*s = s, 3*r*s^3*(s-1) = 1. - _Vaclav Kotesovec_, Nov 18 2023

%t Join[{1}, Table[n! * Sum[k^(n-k) * Binomial[3*n-2*k+1,k] / ((3*n-2*k+1)*(n-k)!), {k,0,n}], {n,1,20}]] (* _Vaclav Kotesovec_, Nov 18 2023 *)

%o (PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*n-2*k+1, k)/((3*n-2*k+1)*(n-k)!));

%Y Cf. A006153, A161633, A364938, A364980.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 15 2023