A370889 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x^2/2)) ).
1, 1, 2, 9, 72, 735, 9000, 133035, 2325120, 46631025, 1053108000, 26484495345, 734652737280, 22280390827695, 733335188826240, 26035824337798275, 991872319953715200, 40360728513989909025, 1747119524427614937600, 80166580022376802179225
Offset: 0
Keywords
Programs
-
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x*exp(x^2/2)))/x)) -
PARI
a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n+1, n-2*k)/(2^k*k!))/(n+1);
Formula
a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n+1,n-2*k)/(2^k * k!).
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