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%I #11 Aug 06 2021 04:33:31
%S 1,5,29,186,1374,11352,105048,1070640,11978640,145558080,1914027840,
%T 27035890560,408891369600,6585851059200,112656894336000,
%U 2038285492992000,38915729475840000,781515776369664000,16475855040820224000,363685261902133248000,8391522945839007744000
%N E.g.f.: log(1 + x) / (1 - x)^3.
%F a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * binomial(n-k+2,2) / k.
%F a(n) ~ log(2) * n^2 * n! / 2. - _Vaclav Kotesovec_, Aug 06 2021
%t nmax = 21; CoefficientList[Series[Log[1 + x]/(1 - x)^3, {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t Table[n! Sum[(-1)^(k + 1) Binomial[n - k + 2, 2]/k , {k, 1, n}], {n, 1, 21}]
%t Table[n!*(((-1)^n*(2*n + 5) - 4*n - 5)/8 + (n+1)*(n+2)*(Log[2] - (-1)^n * LerchPhi[-1, 1, 1 + n])/2), {n, 1, 21}] // Simplify (* _Vaclav Kotesovec_, Aug 06 2021 *)
%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(log(1+x)/(1-x)^3)) \\ _Michel Marcus_, Aug 06 2021
%Y Cf. A000217, A001711, A024167, A109792, A346846, A346847.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Aug 05 2021