A354309 Expansion of e.g.f. 1/(1 - 2*x)^(x/2).
1, 0, 2, 6, 44, 360, 3744, 46200, 662864, 10838016, 198943200, 4050937440, 90613710912, 2208677328000, 58265734055424, 1653914478303360, 50263564166365440, 1628300694034022400, 56012708047907510784, 2039053421375533094400, 78314004507947110456320
Offset: 0
Keywords
Programs
-
Mathematica
With[{nn=20},CoefficientList[Series[1/(1-2x)^(x/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 10 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x)^(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, j*2^(j-2)/(j-1)*v[i-j+1]/(i-j)!)); v;
-
PARI
a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*abs(stirling(n-k, k, 1))/(n-k)!);
Formula
a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * 2^(k-2)/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * |Stirling1(n-k,k)|/(n-k)!.
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n)). - Vaclav Kotesovec, Mar 14 2024