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A108078 Determinant of a Hankel matrix with factorial elements.

%I A108078 #47 Feb 16 2025 08:32:58
%S A108078 1,2,12,576,414720,7166361600,4334215495680000,
%T A108078 125824009525788672000000,230121443546659694208614400000000,
%U A108078 33669808475874225917238947767910400000000000,487707458060712424140716248549520230160793600000000000000
%N A108078 Determinant of a Hankel matrix with factorial elements.
%C A108078 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.
%D A108078 M. J. C. Gover, "The Explicit Inverse of Factorial Hankel Matrices", Department of Mathematics, University of Bradford, 1993
%H A108078 Alois P. Heinz, <a href="/A108078/b108078.txt">Table of n, a(n) for n = 0..32</a>
%H A108078 IPJFACT, <a href="https://web.archive.org/web/20060920043138/http://www.csit.fsu.edu/~burkardt/m_src/test_matrix/ipjfact.m">IPJFACT</a>.
%H A108078 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>
%H A108078 Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>
%F A108078 a(n) = (n+1)! * Product_{i=1..n-1} (i+1)! * (n-i)!.
%F A108078 a(n) = A059332(n)*(n+1)!.
%F A108078 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
%F A108078 a(n) = G(n+1) * G(n+3), where G(n) is the Barnes G function. - _Jan Mangaldan_, May 22 2016
%p A108078 with(LinearAlgebra):
%p A108078 a:= n-> Determinant(Matrix(n, (i,j)->(i+j)!)):
%p A108078 seq(a(n), n=0..10);  # _Alois P. Heinz_, Dec 05 2015
%p A108078 # second Maple program:
%p A108078 a:= n-> (n+1)! * mul((i+1)!*(n-i)!, i=1..n-1):
%p A108078 seq(a(n), n=0..10);  # _Alois P. Heinz_, Dec 05 2015
%t A108078 A108078[n_]:=Det[Table[(i+j)!,{i,1,n},{j,1,n}]]; Array[A108078, 20] (* _Enrique Pérez Herrero_, May 20 2011 *)
%t A108078 Table[BarnesG[n + 1] BarnesG[n + 3], {n, 20}] (* _Jan Mangaldan_, May 22 2016 *)
%o A108078 (MATLAB)
%o A108078 % the sequence is easily made by:
%o A108078 for i=1:n det(gallery('ipjfact',i,0)) end
%o A108078 % or, more explicitly, by:
%o A108078 d = 1; for i=1:n-1 d = d*factorial(i+1)*factorial(n-i); end d = d*factorial(n+1);
%Y A108078 Cf. A000178, A002514, A024356, A056886, A056887, A059332, A293707.
%K A108078 easy,nonn
%O A108078 0,2
%A A108078 _Paul Max Payton_, Jun 03 2005
%E A108078 a(0)=1 prepended and some terms corrected by _Alois P. Heinz_, Dec 05 2015