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 A091530 #35 Apr 09 2025 11:21:01
%S A091530 1,4,16,73,388,2396,17024,137544,1248816,12603288,140018688,
%T A091530 1698063552,22318009344,315942698880,4791898275840,77510315197440,
%U A091530 1331759355586560,24220225133061120,464796175236710400,9385769913543475200,198936154022512435200,4415822707430415052800
%N A091530 a(n) = Sum_{k=1..n} H(k) k! (n-k)!, where H(k) is the k-th harmonic number.
%H A091530 Vladimir Kruchinin, Dmitry Kruchinin, and Yuriy Shablya, <a href="https://mtjpamjournal.com/papers/article_id_mtjpam-d-24-00045/">Some identities for harmonic numbers based on composition of generating functions</a>, Montes Taurus J. Pure Appl. Math. (2024) Vol. 6, No. 3, 363-374. See p. 2.
%F A091530 E.g.f.: (-li[2]((2-x)*x) + 2*li[2](x) - 2*log(1-x)*(x-1)) / (x-2)^2 + (log(1-x) * (4*x^2 + log(1-x)*x - 14*x - log(1-x) + 12)) / (2*(x-2)^2*(x-1)) where li[2](x) is the dilogarithm of x. - _Vladimir Kruchinin_, Jan 03 2024
%F A091530 a(n) ~ n! * (log(n) + gamma), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 03 2024
%p A091530 H:= proc(n) H(n):= `if`(n=0, 0, H(n-1)+1/n) end:
%p A091530 a:= n-> add(H(k)*k!*(n-k)!, k=1..n):
%p A091530 seq(a(n), n=1..22); # _Alois P. Heinz_, Jan 03 2024
%t A091530 Table[ Sum[ HarmonicNumber[k]k!(n - k)!, {k, 1, n}], {n, 1, 20}] (* _Robert G. Wilson v_, Jan 14 2004 *)
%o A091530 (PARI) a(n) = sum(k=1, n, sum(i=1, k, 1/i)*k!*(n-k)!); \\ _Michel Marcus_, Jan 03 2024
%Y A091530 Cf. A000142, A001008, A002805, A091528.
%K A091530 nonn
%O A091530 1,2
%A A091530 _Leroy Quet_, Jan 08 2004
%E A091530 Extended by _Robert G. Wilson v_, Jan 14 2004