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a(n) = n! * Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * H(k), where H(k) is the k-th harmonic number.
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%I #9 Jun 04 2022 06:41:31

%S 0,1,5,50,854,22354,833244,41974176,2748169584,226916044848,

%T 23069499189120,2831994888419520,413051278946186880,

%U 70608112721914654080,13982696139441640584960,3175762393024883382067200,820007850688478572529203200,238863690100874514528150681600

%N a(n) = n! * Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * H(k), where H(k) is the k-th harmonic number.

%H Vaclav Kotesovec, <a href="/A354685/b354685.txt">Table of n, a(n) for n = 0..250</a>

%H Vaclav Kotesovec, <a href="/A354685/a354685.jpg">Graph - the asymptotic ratio</a>

%F Sum_{n>=0} a(n) * x^n / n!^2 = Sum_{n>=1} H(n) * (-log(1-x))^n / n!.

%F a(n) ~ n!^2 * (log(log(n)) + gamma + 1/log(n)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jun 03 2022

%t Table[n! Sum[(-1)^(n - k) StirlingS1[n, k] HarmonicNumber[k], {k, 1, n}], {n, 0, 17}]

%t nmax = 17; CoefficientList[Series[Sum[HarmonicNumber[k] (-Log[1 - x])^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2

%Y Cf. A001008, A002805, A087751, A222059, A302548, A354686.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jun 03 2022