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%I #30 Oct 28 2022 09:55:19
%S 1,1,2,3,24,20,720,630,4480,36288,3628800,3326400,479001600,444787200,
%T 5811886080,81729648000,20922789888000,19760412672000,
%U 6402373705728000,6082255020441600,115852476579840000,2322315553259520000
%N a(n) = numerator(n!/n^2).
%C Numerator of expansion of dilog(x) = Li_2(x) = -Integral_{t=0..x} (log(1-t)/t)*dt. See the Weisstein link.
%C E.g.f. of {a(n)/A014973(n)}_{n>=1} is Li_2(x) (with 0 for n=0).
%H Vincenzo Librandi, <a href="/A092043/b092043.txt">Table of n, a(n) for n = 1..200</a>
%H A. N. Kirillov, <a href="http://arXiv.org/abs/hep-th/9408113">Dilogarithm identities</a>, arXiv:hep-th/9408113, 1994.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>
%F From _Wolfdieter Lang_, Apr 28 2017: (Start)
%F a(n) = numerator(n!/n^2) = numerator((n-1)!/n), n >= 1. See the name.
%F E.g.f. {a(n)/A014973(n)}_{n>=1} with 0 for n=0 is Li_2(x). See the comment.
%F (-1)^n*a(n+1)/A014973(n+1) = (-1)^n*n!/(n+1) = Sum_{k=0..n} Stirling1(n, k)*Bernoulli(k), with Stirling1 = A048994 and Bernoulli(k) = A027641(k)/A027642(k), n >= 0. From inverting the formula for B(k) in terms of Stirling2 = A048993.(End)
%F From _Wolfdieter Lang_, Oct 26 2022: (Start)
%F a(n) = (n-1)!/gcd(n,(n-1)!) = A000142(n-1)/A181569(n-1), n >= 1.
%F The expansion of (1+x)*exp(x) has coefficients A014973(n+1)/a(n+1), for n >= 0. (End)
%t Table[Numerator[n!/n^2], {n, 1, 40}] (* _Vincenzo Librandi_, Apr 15 2014 *)
%t Table[(n-1)!/n,{n,30}]//Numerator (* _Harvey P. Dale_, Apr 03 2018 *)
%o (PARI) a(n)=numerator(n!/n^2)
%o (PARI) a(n)=numerator(polcoeff(serlaplace(dilog(x)),n))
%o (Magma) [Numerator(Factorial(n)/n^2): n in [1..30]]; // _Vincenzo Librandi_, Apr 15 2014
%Y Denominator is in A014973.
%Y Cf. A000142, A001819, A181569.
%K nonn,easy,frac
%O 1,3
%A _Ralf Stephan_, Mar 28 2004
%E Comment rewritten by _Wolfdieter Lang_, Apr 28 2017
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