A108078 Determinant of a Hankel matrix with factorial elements.

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

Offset

0,2

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

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

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 *)

Prog

(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);

Crossrefs

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

Sequence in context: A050643 A145513 A002860 * A052129 A216335 A173104

Adjacent sequences: A108075 A108076 A108077 * A108079 A108080 A108081

Keyword

easy,nonn

Author

Paul Max Payton, Jun 03 2005

Extensions

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

Status

approved