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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 (list; graph; refs; listen; history; text; internal format)
 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,changed 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

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Last modified July 30 07:41 EDT 2021. Contains 346348 sequences. (Running on oeis4.)