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A351493 Expansion of e.g.f. (1 - x)^(-x^3/6).

%I A351493 #25 May 12 2022 15:18:56
%S A351493 1,0,0,0,4,10,40,210,1904,15120,132600,1293600,14303520,171531360,
%T A351493 2223464880,31023392400,464541960960,7424367350400,126124766476800,
%U A351493 2269425252931200,43119553374460800,862673918061715200,18126931548822835200,399119899456951411200
%N A351493 Expansion of e.g.f. (1 - x)^(-x^3/6).
%H A351493 Seiichi Manyama, <a href="/A351493/b351493.txt">Table of n, a(n) for n = 0..451</a>
%F A351493 a(0) = 1; a(n) = (n-1)!/6 * Sum_{k=4..n} k/(k-3) * a(n-k)/(n-k)!.
%F A351493 a(n) = n! * Sum_{k=0..floor(n/4)} |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
%F A351493 a(n) ~ sqrt(2*Pi) * n^(n - 1/3) / (Gamma(1/6) * exp(n)). - _Vaclav Kotesovec_, May 04 2022
%o A351493 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^3/6)))
%o A351493 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^3/6*log(1-x))))
%o A351493 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=4, i, j/(j-3)*v[i-j+1]/(i-j)!)); v;
%o A351493 (PARI) a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));
%Y A351493 Cf. A351492, A351506, A353229.
%K A351493 nonn
%O A351493 0,5
%A A351493 _Seiichi Manyama_, May 02 2022