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%I #31 Jan 03 2024 11:10:45
%S 1,4,16,73,388,2396,17024,137544,1248816,12603288,140018688,
%T 1698063552,22318009344,315942698880,4791898275840,77510315197440,
%U 1331759355586560,24220225133061120,464796175236710400,9385769913543475200,198936154022512435200,4415822707430415052800
%N a(n) = Sum_{k=1..n} H(k) k! (n-k)!, where H(k) is the k-th harmonic number.
%F 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 a(n) ~ n! * (log(n) + gamma), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 03 2024
%p H:= proc(n) H(n):= `if`(n=0, 0, H(n-1)+1/n) end:
%p a:= n-> add(H(k)*k!*(n-k)!, k=1..n):
%p seq(a(n), n=1..22); # _Alois P. Heinz_, Jan 03 2024
%t Table[ Sum[ HarmonicNumber[k]k!(n - k)!, {k, 1, n}], {n, 1, 20}] (* _Robert G. Wilson v_, Jan 14 2004 *)
%o (PARI) a(n) = sum(k=1, n, sum(i=1, k, 1/i)*k!*(n-k)!); \\ _Michel Marcus_, Jan 03 2024
%Y Cf. A000142, A001008, A002805, A091528.
%K nonn
%O 1,2
%A _Leroy Quet_, Jan 08 2004
%E Extended by _Robert G. Wilson v_, Jan 14 2004
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