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%I #22 Jun 22 2022 14:49:48
%S 1,3,14,70,444,3108,25584,230256,2342880,25771680,312888960,
%T 4067556480,57424792320,861371884800,13869128448000,235775183616000,
%U 4264876094976000,81032645804544000,1627055289796608000,34168161085728768000,754132445894209536000,17345046255566819328000
%N Expansion of e.g.f. log(1+x)/(1-x)^2.
%H G. C. Greubel, <a href="/A109792/b109792.txt">Table of n, a(n) for n = 1..400</a>
%F a(n) = n!*Sum_{k=1..n} Sum_{i=1..k} (-1)^(i+1)/i.
%F a(n) ~ n!*n*log(2). - _Vaclav Kotesovec_, Jun 27 2013
%F a(n) = n!*((-1)^n*(n + 1)*LerchPhi(-1, 1, n + 2) + log(2)*(n + 1) + ((-1)^(n + 1) - 1) / 2). - _Peter Luschny_, Jun 22 2022
%t CoefficientList[Series[Log[1+x]/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 27 2013 *)
%t a[n_] := n! ((-1)^n (n + 1) LerchPhi[-1, 1, n + 2] + Log[2] (n + 1) + ((-1)^(n + 1) - 1) /2); Table[Simplify[a[n]], {n, 1, 22}] (* _Peter Luschny_, Jun 22 2022 *)
%o (PARI) for(n=1,25, print1(n!*sum(k=1,n, sum(i=1, k, (-1)^(i+1)/i)), ", ")) \\ _G. C. Greubel_, Jan 21 2017
%Y Cf. A001705, A092692.
%K easy,nonn
%O 1,2
%A _Vladeta Jovovic_, Aug 14 2005
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