A353222 Expansion of e.g.f. (1 - x^3)^(-1/x).
1, 0, 2, 0, 12, 60, 120, 2520, 15120, 90720, 1693440, 13305600, 140374080, 2724321600, 27744837120, 414096883200, 8689288608000, 111399326438400, 2114134793971200, 48501156601497600, 759659036405068800, 17279306372135808000, 434100706059205785600
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)^(-1/x))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^3)/x))) -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, (i+1)\3, (3*j-1)/j*v[i-3*j+2]/(i-3*j+1)!)); v;
Formula
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor((n+1)/3)} (3*k-1)/k * a(n-3*k+1)/(n-3*k+1)!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3*exp(n)). - Vaclav Kotesovec, May 04 2022