A354319 - OEIS

%I #15 Mar 14 2024 09:03:58

%S 1,0,1,3,19,150,1497,17955,251681,4036284,72874125,1462571055,

%T 32297755803,778188449610,20313917363733,571081958323695,

%U 17201321168216385,552635193533958360,18863471310967732473,681711909339186154395,26003437607893415476995

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

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

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

%F a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 3/8) / (Gamma(1/8) * exp(n)). - _Vaclav Kotesovec_, Mar 14 2024

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

%o (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;

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

%Y Cf. A354309, A354323, A354327.

%K nonn

%O 0,4

%A _Seiichi Manyama_, May 24 2022