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%I #31 Mar 04 2023 15:13:40
%S 1,0,0,6,36,210,2070,24864,310632,4337544,68922360,1205002656,
%T 22844264256,469287123552,10397824478496,246800350393344,
%U 6246190572981120,167972669001740160,4783274802508890240,143775432034543203840,4548946867429143444480
%N Expansion of e.g.f. 1/(1 + log(1 - x)^3).
%H Seiichi Manyama, <a href="/A353118/b353118.txt">Table of n, a(n) for n = 0..418</a>
%F a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).
%F a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|.
%F a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3 * (exp(1) - 1)^(n+1)). - _Vaclav Kotesovec_, May 07 2022
%t With[{nn=20},CoefficientList[Series[1/(1+Log[1-x]^3),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 04 2023 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)^3)))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i, j)*abs(stirling(j, 3, 1))*v[i-j+1])); v;
%o (PARI) a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1)));
%Y Cf. A052811, A353119, A353200.
%Y Cf. A346922.
%K nonn
%O 0,4
%A _Seiichi Manyama_, May 06 2022
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