A353226 Expansion of e.g.f. (1 - x^2)^(-x).
1, 0, 0, 6, 0, 60, 360, 1680, 20160, 151200, 1663200, 17962560, 219542400, 2854051200, 40441040640, 606356150400, 9793028044800, 166481476761600, 3017626733721600, 57359043873331200, 1153275200453376000, 24233844054131712000, 535361100608439705600
Offset: 0
Keywords
Programs
-
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^2)^(-x))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^2)))) -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, (i+1)\2, (2*j-1)/(j-1)*v[i-2*j+2]/(i-2*j+1)!)); v;
-
PARI
a(n) = n!*sum(k=0, n\2, abs(stirling(k, n-2*k, 1))/k!);
Formula
a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+1)/2)} (2*k-1)/(k-1) * a(n-2*k+1)/(n-2*k+1)!.
a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(k,n-2*k)|/k!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (2*exp(n)). - Vaclav Kotesovec, May 04 2022