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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).

%I A079478 #37 Dec 04 2023 03:42:24
%S A079478 1,2,72,172800,60963840000,5574884681318400000,
%T A079478 205619158526859285626880000000,
%U A079478 4394314874750658447092552646524928000000000,73955304765761130113502867875624106401967636480000000000000
%N 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).
%C A079478 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 A079478 Alois P. Heinz, <a href="/A079478/b079478.txt">Table of n, a(n) for n = 0..20</a>
%F A079478 a(n) = (n+1)*(Product_{i=0..n} (n+i)!)/Product_{i=1..n+1} i!.
%F A079478 a(n) = A000178(2n)/A000178(n)^2, i.e., "central supercombinations" by analogy with A000984. - _Henry Bottomley_, May 14 2005
%F A079478 a(n) = Product_{j=1..n} Product_{i=1..n} (i + j). - _Alexander Adamchuk_, Apr 12 2006
%F A079478 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 A079478 a(n) = BarnesG(2*n+2) / BarnesG(n+2)^2. - _Vaclav Kotesovec_, Feb 28 2019
%F A079478 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 A079478 Determinant of M(2) is 1/(x^4 + 12*x^3 + 53*x^2 + 102*x + 72) hence a(2)=72.
%p A079478 seq(mul(mul(k+j,j=1..n), k=1..n), n=0..8); # _Zerinvary Lajos_, Jun 01 2007
%t A079478 Table[Product[Product[(i+j),{i,1,n}],{j,1,n}],{n,0,10}] (* _Alexander Adamchuk_, Apr 12 2006 *)
%t A079478 Table[BarnesG[2*n+2] / BarnesG[n+2]^2, {n,0,10}] (* _Vaclav Kotesovec_, Feb 28 2019 *)
%o A079478 (PARI) a(n)=(n+1)*prod(i=0,n,(n+i)!)/prod(i=1,n+1,i!)
%o A079478 (PARI) a(n) = prod(i=1, n, prod(j=1, n, i+j)); \\ _Michel Marcus_, Feb 27 2019
%o A079478 (Python)
%o A079478 from math import prod, factorial
%o A079478 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 A079478 Cf. A011379.
%Y A079478 Central column in triangle A009963.
%K A079478 nonn
%O A079478 0,2
%A A079478 _Benoit Cloitre_, Jan 15 2003