A372251 - OEIS

%I #12 Apr 24 2024 08:20:42

%S 1,1,6,73,1364,34586,1110496,43207004,1976199792,103925934712,

%T 6178846168976,409847155094840,30007066358487040,2403751529017358144,

%U 209131503815967330816,19637892118783264231936,1979605910448187576510208,213226210180592877512104832

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

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

%F a(n) ~ sqrt((1 + s)/(4 + 9*s)) * s^(2*n + 1) * (2 + 3*s)^n * n^(n-1) / (2^n * exp(n)), where s = 1.470103625022272111740158699814771551850270522048... is the root of the equation log(s) = (1 + s)/(2 + 3*s). - _Vaclav Kotesovec_, Apr 24 2024

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

%Y Cf. A138860, A372246.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 24 2024