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A079478

Coefficient of x^0 in P(n,x) = (Product_{i=0..n-1} i!^2)/matdet(M(n)) of degree n^2 where M(n) is the n X n matrix m(i,j) = 1/(i+j+x).

1, 2, 72, 172800, 60963840000, 5574884681318400000, 205619158526859285626880000000, 4394314874750658447092552646524928000000000, 73955304765761130113502867875624106401967636480000000000000

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OFFSET

0,2

COMMENTS

Product of all matrix elements of n X n matrix M(i,j) = i+j (i,j=1..n). - Alexander Adamchuk, Apr 12 2006

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..20

FORMULA

a(n) = (n+1)*(Product_{i=0..n} (n+i)!)/Product_{i=1..n+1} i!.

a(n) = A000178(2n)/A000178(n)^2, i.e., "central supercombinations" by analogy with A000984. - Henry Bottomley, May 14 2005

a(n) = Product_{j=1..n} Product_{i=1..n} (i + j). - Alexander Adamchuk, Apr 12 2006

Asymptotic: a(n) ~ (2*n+1)^(2*n^2 + 2*n + 5/12)*(n+1)^(-n^2 - 2*n - 5/6) * exp(-zeta'(-1) - (3/2)*n^2 + 3/4)/(sqrt(2*Pi)). - Peter Luschny, Nov 26 2012

a(n) = BarnesG(2*n+2) / BarnesG(n+2)^2. - Vaclav Kotesovec, Feb 28 2019

a(n) ~ A * 2^(2*n*(n+1) - 1/12) * n^(n^2 - 5/12) / (sqrt(Pi) * exp(3*n^2/2 + 1/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Dec 04 2023

EXAMPLE

Determinant of M(2) is 1/(x^4 + 12*x^3 + 53*x^2 + 102*x + 72) hence a(2)=72.

MAPLE

seq(mul(mul(k+j, j=1..n), k=1..n), n=0..8); # Zerinvary Lajos, Jun 01 2007

MATHEMATICA

Table[Product[Product[(i+j), {i, 1, n}], {j, 1, n}], {n, 0, 10}] (* Alexander Adamchuk, Apr 12 2006 *)

Table[BarnesG[2*n+2] / BarnesG[n+2]^2, {n, 0, 10}] (* Vaclav Kotesovec, Feb 28 2019 *)

PROG

(PARI) a(n)=(n+1)*prod(i=0, n, (n+i)!)/prod(i=1, n+1, i!)

(PARI) a(n) = prod(i=1, n, prod(j=1, n, i+j)); \\ Michel Marcus, Feb 27 2019

(Python)

from math import prod, factorial