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A108078
Determinant of a Hankel matrix with factorial elements.
2
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
IPJFACT, IPJFACT.
Eric Weisstein's World of Mathematics, 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);
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