A364983 - OEIS

%I #16 Nov 11 2024 13:01:02

%S 1,1,8,111,2332,66125,2368086,102616759,5222638856,305436798009,

%T 20186656927210,1488021110087171,121044207712073196,

%U 10771321471267219525,1040877104088653696606,108549742436141933697135,12151467262433697322437136,1453367472748861203540942065

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

%F a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (3*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A001764(k)/(n-k)!.

%F a(n) ~ sqrt(3) * sqrt(1 + LambertW(4/27)) * n^(n-1) / (2^(3/2) * exp(n) * LambertW(4/27)^n). - _Vaclav Kotesovec_, Nov 11 2024

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

%Y Cf. A364984, A364985, A364986.

%Y Cf. A006153, A295238, A364987.

%Y Cf. A001764, A307678.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 15 2023