A354320 Expansion of e.g.f. 1/(1 - 4*x)^(x/8).
1, 0, 1, 6, 67, 1020, 19767, 464310, 12802121, 405017928, 14454250785, 574259123790, 25131727031163, 1201109694719220, 62238037299307863, 3475264183358721390, 208017790077615619665, 13286691367919839674000, 901996048369381319539713
Offset: 0
Keywords
Crossrefs
Cf. A354328.
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-4*x)^(x/8))) -
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*4^(j-2)/(j-1)*v[i-j+1]/(i-j)!)/2); v;
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PARI
a(n) = n!*sum(k=0, n\2, 4^(n-2*k)*abs(stirling(n-k, k, 1))/(2^k*(n-k)!));
Formula
a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=2..n} k * 4^(k-2)/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 4^(n-2*k) * |Stirling1(n-k,k)|/(2^k * (n-k)!).