A351156 Expansion of e.g.f. (1 - x^3/6)^(-x).
1, 0, 0, 0, 4, 0, 0, 70, 560, 0, 5600, 92400, 369600, 1201200, 30830800, 252252000, 1210809600, 19059040000, 240143904000, 1738184448000, 22451549120000, 342205063200000, 3417705170880000, 43866126368064000, 732641268463104000, 9234973972224000000
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3/6)^(-x))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^3/6)))) -
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+2)\3, (3*j-2)/((j-1)*6^(j-1))*v[i-3*j+3]/(i-3*j+2)!)); v;
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PARI
a(n) = n!*sum(k=0, n\3, abs(stirling(k, n-3*k, 1))/(6^k*k!));
Formula
a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+2)/3)} (3*k-2)/((k-1) * 6^(k-1)) * a(n-3*k+2)/(n-3*k+2)!.
a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(k,n-3*k)|/(6^k*k!).
a(n) ~ sqrt(2*Pi) * n^(n - 1/2 + 6^(1/3)) / (Gamma(6^(1/3)) * 3^(6^(1/3)) * exp(n) * 6^(n/3)). - Vaclav Kotesovec, May 04 2022