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Expansion of e.g.f. 1/(1 + (exp(x) - 1)^4 / 24).
5

%I #16 Feb 26 2023 12:54:54

%S 1,0,0,0,-1,-10,-65,-350,-1631,-5250,18395,685850,10485739,127737610,

%T 1336804105,11432407350,54280609109,-712071643930,-29671691715185,

%U -660215774400350,-11770593620859521,-176475952496559870,-2055362595355830475,-9749893741512339250

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

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

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(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, j)*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);

%Y Cf. A354391, A354392, A354394.

%Y Cf. A346895, A354390, A354397.

%K sign

%O 0,6

%A _Seiichi Manyama_, May 25 2022