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A354398 Expansion of e.g.f. exp( -(exp(x) - 1)^5 / 120 ).

%I A354398 #15 Jul 23 2025 19:50:37
%S A354398 1,0,0,0,0,-1,-15,-140,-1050,-6951,-42399,-239800,-1164570,-2553551,
%T A354398 54771717,1384600854,23301803070,340911045929,4600861076433,
%U A354398 58236569430172,687816515641206,7315220762286129,61629305427537309,140107851269900954,-11001310744922517426
%N A354398 Expansion of e.g.f. exp( -(exp(x) - 1)^5 / 120 ).
%F A354398 a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,5) * a(n-k).
%F A354398 a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/((-120)^k * k!).
%t A354398 With[{nn=30},CoefficientList[Series[Exp[-(Exp[x]-1)^5/120],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 23 2025 *)
%o A354398 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(exp(x)-1)^5/120)))
%o A354398 (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;
%o A354398 (PARI) a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/((-120)^k*k!));
%Y A354398 Cf. A000587, A354395, A354396, A354397.
%Y A354398 Cf. A327506, A346977, A354394.
%K A354398 sign
%O A354398 0,7
%A A354398 _Seiichi Manyama_, May 25 2022