A354326 Expansion of e.g.f. 1/(1 - x/8 * (exp(4 * x) - 1)).
1, 0, 1, 6, 38, 280, 2538, 27524, 341912, 4754880, 73322360, 1244282512, 23048700912, 462565343552, 9996300546512, 231444311970720, 5715911385442688, 149988948332148736, 4167328800543910272, 122218355207805620480, 3773036019063284645120
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=20},CoefficientList[Series[1/(1-x/8 (Exp[4x]-1)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 03 2023 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/8*(exp(4*x)-1)))) -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*4^(j-2)*binomial(i, j)*v[i-j+1])/2); v;
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PARI
a(n) = n!*sum(k=0, n\2, 4^(n-2*k)*k!*stirling(n-k, k, 2)/(2^k*(n-k)!));
Formula
a(0) = 1; a(n) = (1/2) * Sum_{k=2..n} k * 4^(k-2) * binomial(n,k) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 4^(n-2*k) * k! * Stirling2(n-k,k)/(2^k * (n-k)!).