This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354685 #9 Jun 04 2022 06:41:31
%S A354685 0,1,5,50,854,22354,833244,41974176,2748169584,226916044848,
%T A354685 23069499189120,2831994888419520,413051278946186880,
%U A354685 70608112721914654080,13982696139441640584960,3175762393024883382067200,820007850688478572529203200,238863690100874514528150681600
%N A354685 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 A354685 Vaclav Kotesovec, <a href="/A354685/b354685.txt">Table of n, a(n) for n = 0..250</a>
%H A354685 Vaclav Kotesovec, <a href="/A354685/a354685.jpg">Graph - the asymptotic ratio</a>
%F A354685 Sum_{n>=0} a(n) * x^n / n!^2 = Sum_{n>=1} H(n) * (-log(1-x))^n / n!.
%F A354685 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 A354685 Table[n! Sum[(-1)^(n - k) StirlingS1[n, k] HarmonicNumber[k], {k, 1, n}], {n, 0, 17}]
%t A354685 nmax = 17; CoefficientList[Series[Sum[HarmonicNumber[k] (-Log[1 - x])^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y A354685 Cf. A001008, A002805, A087751, A222059, A302548, A354686.
%K A354685 nonn
%O A354685 0,3
%A A354685 _Ilya Gutkovskiy_, Jun 03 2022