A108078 Determinant of a Hankel matrix with factorial elements.

1, 2, 12, 576, 414720, 7166361600, 4334215495680000, 125824009525788672000000, 230121443546659694208614400000000, 33669808475874225917238947767910400000000000, 487707458060712424140716248549520230160793600000000000000

Offset: 0

Author

Paul Max Payton, Jun 03 2005

Keywords

• easy
• nonn

Comments

The term (n=1) is a degenerate case, a matrix with single element 2. This sequence involves products of binomial coefficients and is related to the superfactorial function.

References

• M. J. C. Gover, "The Explicit Inverse of Factorial Hankel Matrices", Department of Mathematics, University of Bradford, 1993

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..32
• IPJFACT, IPJFACT.
• Eric Weisstein's World of Mathematics, Barnes G-Function
• Wikipedia, Barnes G-function

Crossrefs

Cf. A000178, A002514, A024356, A056886, A056887, A059332, A293707.

Programs

MATLAB

% the sequence is easily made by:
for i=1:n det(gallery('ipjfact',i,0)) end
% or, more explicitly, by:
d = 1; for i=1:n-1 d = d*factorial(i+1)*factorial(n-i); end d = d*factorial(n+1);

Maple

with(LinearAlgebra):
a:= n-> Determinant(Matrix(n, (i,j)->(i+j)!)):
seq(a(n), n=0..10);  # Alois P. Heinz, Dec 05 2015
# second Maple program:
a:= n-> (n+1)! * mul((i+1)!*(n-i)!, i=1..n-1):
seq(a(n), n=0..10);  # Alois P. Heinz, Dec 05 2015

Mathematica

A108078[n_]:=Det[Table[(i+j)!,{i,1,n},{j,1,n}]]; Array[A108078, 20] (* Enrique Pérez Herrero, May 20 2011 *)
Table[BarnesG[n + 1] BarnesG[n + 3], {n, 20}] (* Jan Mangaldan, May 22 2016 *)

Formula

a(n) = (n+1)! * Product_{i=1..n-1} (i+1)! * (n-i)!.

a(n) = A059332(n)*(n+1)!.

a(n) ~ n^(n^2 + 2*n + 11/6) * 2^(n+1) * Pi^(n+1) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 16 2016

a(n) = G(n+1) * G(n+3), where G(n) is the Barnes G function. - Jan Mangaldan, May 22 2016

Extensions

a(0)=1 prepended and some terms corrected by Alois P. Heinz, Dec 05 2015