A354319 Expansion of e.g.f. 1/(1 - 2*x)^(x/4).

1, 0, 1, 3, 19, 150, 1497, 17955, 251681, 4036284, 72874125, 1462571055, 32297755803, 778188449610, 20313917363733, 571081958323695, 17201321168216385, 552635193533958360, 18863471310967732473, 681711909339186154395, 26003437607893415476995

Offset: 0

Author

Seiichi Manyama, May 24 2022

Keywords

• nonn

Crossrefs

Cf. A354309, A354323, A354327.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x)^(x/4)))

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-3)/(j-1)*v[i-j+1]/(i-j)!)); v;

PARI

a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * 2^(k-3)/(k-1) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-3*k) * |Stirling1(n-k,k)|/(n-k)!.

a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 3/8) / (Gamma(1/8) * exp(n)). - Vaclav Kotesovec, Mar 14 2024