A092043 - OEIS

%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