A375721 - OEIS

%I #13 Sep 06 2024 06:12:28

%S 1,6,60,822,14238,297684,7286076,204251328,6450932448,226613038608,

%T 8763294140064,369900822475728,16922169163019088,833991953707934496,

%U 44050579327333028448,2482381132145285334912,148660444826262311114880,9427874254540824544312320

%N Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^2.

%F a(n) = Sum_{k=0..n} 3^k * (k+1)! * |Stirling1(n,k)|.

%F a(0) = 1; a(n) = 3 * Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).

%F a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / (9 * exp(2*n/3) * (exp(1/3) - 1)^(n+2)). - _Vaclav Kotesovec_, Sep 06 2024

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))^2))

%o (PARI) a(n) = sum(k=0, n, 3^k*(k+1)!*abs(stirling(n, k, 1)));

%Y Cf. A354263, A375722.

%Y Cf. A052801, A367474.

%Y Cf. A367472.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 25 2024