A354398 Expansion of e.g.f. exp( -(exp(x) - 1)^5 / 120 ).
1, 0, 0, 0, 0, -1, -15, -140, -1050, -6951, -42399, -239800, -1164570, -2553551, 54771717, 1384600854, 23301803070, 340911045929, 4600861076433, 58236569430172, 687816515641206, 7315220762286129, 61629305427537309, 140107851269900954, -11001310744922517426
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=30},CoefficientList[Series[Exp[-(Exp[x]-1)^5/120],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 23 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(exp(x)-1)^5/120))) -
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, 5, 2)*v[i-j+1])); v;
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PARI
a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/((-120)^k*k!));
Formula
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/((-120)^k * k!).