A360474 - OEIS

%I #12 Feb 17 2023 16:02:50

%S 1,1,7,94,1921,53036,1849789,78070462,3869909537,220427550712,

%T 14188370562901,1018570771664546,80692202644742737,

%U 6992855583524143204,658076908751441373965,66833181471569822199886,7285736943975575120653249,848589321771735983890457072

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

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

%F a(n) ~ 2^(n - 1/2) * n^(n-1) * s^(2*n + 1) * log(s)^(n + 1/2) / (sqrt(1 + 2*log(s) - 4*log(s)^2) * exp(n) * (1 - 2*log(s))^n), where s = 1.473428520956658037187728756446912746332041803082... is the root of the equation 2*log(s)*(1 + LambertW(log(s))) = 1. - _Vaclav Kotesovec_, Feb 17 2023

%t Join[{1}, Table[Sum[k^(n-k) * (2*n+1)^(k-1) * Binomial[n,k], {k,0,n}], {n,1,20}]] (* _Vaclav Kotesovec_, Feb 17 2023 *)

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

%Y Cf. A162695, A360473.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 08 2023