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%I #32 Sep 07 2023 03:01:46
%S 1,1,1,10,10,604,-1844,107344,-1201400,42193576,-875584376,
%T 29853569008,-880141783184,32865860907424,-1216481572723616,
%U 51296026356128512,-2244334822166729600,106984479644794783360,-5358207684820194270080,286466413246622566048000
%N Expansion of e.g.f. 1/(1 - log(1 + 3*x))^(1/3).
%F a(n) = Sum_{k=0..n} 3^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling1(n,k).
%F For n > 0, a(n) = n!*Sum_{k=1..n} a(n-k)*(2/n/3-1/k)*(-3)^k/(n-k)!. - _Tani Akinari_, Sep 07 2023
%F a(n) ~ -(-1)^n * 3^(n-1) * n! / (n * log(n)^(4/3)) * (1 - 4*(1+gamma)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 07 2023
%t m = 19; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 3*x])^(-1/3), {x, 0, m}], x] (* _Amiram Eldar_, Mar 05 2022 *)
%t Table[Sum[3^(n-k) * Product[3*j+1, {j,0,k-1}] * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 07 2023 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+3*x))^(1/3)))
%o (PARI) a(n) = sum(k=0, n, 3^(n-k)*prod(j=0, k-1, 3*j+1)*stirling(n, k, 1));
%o (Maxima) a[n]:=if n=0 then 1 else n!*sum(a[n-k]*(2/n/3-1/k)*(-3)^k/(n-k)!,k,1,n);
%o makelist(a[n],n,0,50); /* _Tani Akinari_, Sep 07 2023 */
%Y Cf. A006252, A097397, A352073.
%Y Cf. A007559, A133480, A352113, A352118.
%K sign
%O 0,4
%A _Seiichi Manyama_, Mar 05 2022
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