A092043 a(n) = numerator(n!/n^2).

1, 1, 2, 3, 24, 20, 720, 630, 4480, 36288, 3628800, 3326400, 479001600, 444787200, 5811886080, 81729648000, 20922789888000, 19760412672000, 6402373705728000, 6082255020441600, 115852476579840000, 2322315553259520000

Offset: 1

Author

Ralf Stephan, Mar 28 2004

Keywords

• nonn
• easy
• frac

Comments

Numerator of expansion of dilog(x) = Li_2(x) = -Integral_{t=0..x} (log(1-t)/t)*dt. See the Weisstein link.

E.g.f. of {a(n)/A014973(n)}_{n>=1} is Li_2(x) (with 0 for n=0).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• A. N. Kirillov, Dilogarithm identities, arXiv:hep-th/9408113, 1994.
• Eric Weisstein's World of Mathematics, Dilogarithm

Crossrefs

Denominator is in A014973.

Cf. A000142, A001819, A181569.

Programs

Magma

[Numerator(Factorial(n)/n^2): n in [1..30]]; // Vincenzo Librandi, Apr 15 2014

Mathematica

Table[Numerator[n!/n^2], {n, 1, 40}] (* Vincenzo Librandi, Apr 15 2014 *)
Table[(n-1)!/n,{n,30}]//Numerator (* Harvey P. Dale, Apr 03 2018 *)

PARI

a(n)=numerator(n!/n^2)

PARI

a(n)=numerator(polcoeff(serlaplace(dilog(x)),n))

Formula

From Wolfdieter Lang, Apr 28 2017: (Start)

a(n) = numerator(n!/n^2) = numerator((n-1)!/n), n >= 1. See the name.

E.g.f. {a(n)/A014973(n)}_{n>=1} with 0 for n=0 is Li_2(x). See the comment.

(-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)

From Wolfdieter Lang, Oct 26 2022: (Start)

a(n) = (n-1)!/gcd(n,(n-1)!) = A000142(n-1)/A181569(n-1), n >= 1.

The expansion of (1+x)*exp(x) has coefficients A014973(n+1)/a(n+1), for n >= 0. (End)

Extensions

Comment rewritten by Wolfdieter Lang, Apr 28 2017