A370889 - OEIS

%I #22 Mar 06 2024 11:28:17

%S 1,1,2,9,72,735,9000,133035,2325120,46631025,1053108000,26484495345,

%T 734652737280,22280390827695,733335188826240,26035824337798275,

%U 991872319953715200,40360728513989909025,1747119524427614937600,80166580022376802179225

%N Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x^2/2)) ).

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

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

%F a(n) ~ (1 + 3*LambertW(1/3))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(1/3)) * 3^(3*n/2 + 2) * exp(n) * LambertW(1/3)^(3*(n+1)/2)). - _Vaclav Kotesovec_, Mar 06 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x*exp(x^2/2)))/x))

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

%Y Cf. A161633, A370926.

%Y Cf. A365283.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 05 2024