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Expansion of e.g.f. 1 / sqrt(1 + x^3 * log(1 - x)).
3

%I #11 Aug 25 2024 09:58:23

%S 1,0,0,0,12,30,120,630,19152,166320,1506600,14968800,313014240,

%T 4864860000,72829607760,1116874558800,23605893400320,495461472105600,

%U 10289649464640000,215706738207542400,5222625647551920000,133507746422859513600,3481696859911699968000

%N Expansion of e.g.f. 1 / sqrt(1 + x^3 * log(1 - x)).

%F a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+3)) * |Stirling1(n-3*k,k)|/(6^k*(n-3k)!).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1+x^3*log(1-x))))

%o (PARI) a(n) = n!*sum(k=0, n\4, prod(j=0, k-1, 6*j+3)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

%Y Cf. A351504, A375699, A375700.

%Y Cf. A375718.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Aug 25 2024