A354396 Expansion of e.g.f. exp( -(exp(x) - 1)^3 / 6 ).
1, 0, 0, -1, -6, -25, -80, -91, 1694, 23155, 206340, 1442969, 6928394, -6507865, -752409840, -12953182971, -160186016906, -1548849362085, -9789241693220, 28359195353489, 2378650585685794, 52832659521004495, 855581150441210600, 10878338100191146749
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=30},CoefficientList[Series[Exp[-(Exp[x]-1)^3/6],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 02 2023 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(exp(x)-1)^3/6))) -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 2)*v[i-j+1])); v;
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PARI
a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/((-6)^k*k!));
Formula
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/((-6)^k * k!).