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 A092043 #31 Feb 16 2025 08:32:52 %S A092043 1,1,2,3,24,20,720,630,4480,36288,3628800,3326400,479001600,444787200, %T A092043 5811886080,81729648000,20922789888000,19760412672000, %U A092043 6402373705728000,6082255020441600,115852476579840000,2322315553259520000 %N A092043 a(n) = numerator(n!/n^2). %C A092043 Numerator of expansion of dilog(x) = Li_2(x) = -Integral_{t=0..x} (log(1-t)/t)*dt. See the Weisstein link. %C A092043 E.g.f. of {a(n)/A014973(n)}_{n>=1} is Li_2(x) (with 0 for n=0). %H A092043 Vincenzo Librandi, <a href="/A092043/b092043.txt">Table of n, a(n) for n = 1..200</a> %H A092043 A. N. Kirillov, <a href="http://arXiv.org/abs/hep-th/9408113">Dilogarithm identities</a>, arXiv:hep-th/9408113, 1994. %H A092043 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a> %F A092043 From _Wolfdieter Lang_, Apr 28 2017: (Start) %F A092043 a(n) = numerator(n!/n^2) = numerator((n-1)!/n), n >= 1. See the name. %F A092043 E.g.f. {a(n)/A014973(n)}_{n>=1} with 0 for n=0 is Li_2(x). See the comment. %F A092043 (-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 A092043 From _Wolfdieter Lang_, Oct 26 2022: (Start) %F A092043 a(n) = (n-1)!/gcd(n,(n-1)!) = A000142(n-1)/A181569(n-1), n >= 1. %F A092043 The expansion of (1+x)*exp(x) has coefficients A014973(n+1)/a(n+1), for n >= 0. (End) %t A092043 Table[Numerator[n!/n^2], {n, 1, 40}] (* _Vincenzo Librandi_, Apr 15 2014 *) %t A092043 Table[(n-1)!/n,{n,30}]//Numerator (* _Harvey P. Dale_, Apr 03 2018 *) %o A092043 (PARI) a(n)=numerator(n!/n^2) %o A092043 (PARI) a(n)=numerator(polcoeff(serlaplace(dilog(x)),n)) %o A092043 (Magma) [Numerator(Factorial(n)/n^2): n in [1..30]]; // _Vincenzo Librandi_, Apr 15 2014 %Y A092043 Denominator is in A014973. %Y A092043 Cf. A000142, A001819, A181569. %K A092043 nonn,easy,frac %O A092043 1,3 %A A092043 _Ralf Stephan_, Mar 28 2004 %E A092043 Comment rewritten by _Wolfdieter Lang_, Apr 28 2017