A364941 - OEIS

%I #11 Nov 18 2023 05:50:10

%S 1,1,9,139,3201,98861,3842653,180342471,9926870145,627296384665,

%T 44766115252821,3561306199330859,312531347680052449,

%U 29994317717748851013,3125271184480991706189,351360521075659460743471,42395667639523579933634817,5464885215245368415146646321

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

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

%F a(n) ~ s^2 * sqrt((1 + r*s)/(1 + 2*r*s^2 - 3*r^2*s^2 + 2*r^3*s^3)) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.1208150626316801846776206051780724146363... and s = 1.505405324736640697527292770220289316454393380356... are real roots of the system of equations exp(r*s^2 / (1 - r*s)^2) = s, 2*r*s^2 = (1 - r*s)^3. - _Vaclav Kotesovec_, Nov 18 2023

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

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

%Y Cf. A361142, A364942.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 14 2023