A354310 Expansion of e.g.f. 1/(1 - 3*x)^(x/3).
1, 0, 2, 9, 84, 990, 14754, 264600, 5549424, 133217784, 3601384200, 108249692760, 3580724721672, 129250420556400, 5055196156459344, 212951257371183240, 9612027759287831040, 462798880374787387200, 23675607840207619145664, 1282413928716141429168000
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x)^(x/3))) -
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*3^(j-2)/(j-1)*v[i-j+1]/(i-j)!)); v;
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PARI
a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*abs(stirling(n-k, k, 1))/(n-k)!);
Formula
a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * 3^(k-2)/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * |Stirling1(n-k,k)|/(n-k)!.