%I #37 Dec 04 2023 03:42:24

%S 1,2,72,172800,60963840000,5574884681318400000,

%T 205619158526859285626880000000,

%U 4394314874750658447092552646524928000000000,73955304765761130113502867875624106401967636480000000000000

%N 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).

%C Product of all matrix elements of n X n matrix M(i,j) = i+j (i,j=1..n). - _Alexander Adamchuk_, Apr 12 2006

%H Alois P. Heinz, <a href="/A079478/b079478.txt">Table of n, a(n) for n = 0..20</a>

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

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

%F a(n) = Product_{j=1..n} Product_{i=1..n} (i + j). - _Alexander Adamchuk_, Apr 12 2006

%F 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

%F a(n) = BarnesG(2*n+2) / BarnesG(n+2)^2. - _Vaclav Kotesovec_, Feb 28 2019

%F 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

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

%p seq(mul(mul(k+j,j=1..n), k=1..n), n=0..8); # _Zerinvary Lajos_, Jun 01 2007

%t Table[Product[Product[(i+j),{i,1,n}],{j,1,n}],{n,0,10}] (* _Alexander Adamchuk_, Apr 12 2006 *)

%t Table[BarnesG[2*n+2] / BarnesG[n+2]^2, {n,0,10}] (* _Vaclav Kotesovec_, Feb 28 2019 *)

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

%o (PARI) a(n) = prod(i=1, n, prod(j=1, n, i+j)); \\ _Michel Marcus_, Feb 27 2019

%o (Python)

%o from math import prod, factorial

%o def A079478(n): return prod(i+j for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)<<n # _Chai Wah Wu_, Nov 26 2023

%Y Cf. A011379.

%Y Central column in triangle A009963.

%K nonn

%O 0,2

%A _Benoit Cloitre_, Jan 15 2003