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Expansion of e.g.f. exp( -(exp(x) - 1)^4 / 24 ).
4

%I #12 May 25 2022 09:15:47

%S 1,0,0,0,-1,-10,-65,-350,-1666,-6510,-7855,270050,4948669,63503440,

%T 702095030,6924754200,58870214129,356043924590,-615569993285,

%U -74306502570650,-1783956267419536,-32695418069393310,-520090808927130925,-7317310078355307250,-87056749651694635451

%N Expansion of e.g.f. exp( -(exp(x) - 1)^4 / 24 ).

%F a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,4) * a(n-k).

%F a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/((-24)^k * k!).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(exp(x)-1)^4/24)))

%o (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, 4, 2)*v[i-j+1])); v;

%o (PARI) a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/((-24)^k*k!));

%Y Cf. A000587, A354395, A354396, A354398.

%Y Cf. A327505, A354393.

%K sign

%O 0,6

%A _Seiichi Manyama_, May 25 2022