|
|
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).
|
|
36
|
|
|
1, 2, 72, 172800, 60963840000, 5574884681318400000, 205619158526859285626880000000, 4394314874750658447092552646524928000000000, 73955304765761130113502867875624106401967636480000000000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
a(n) = (n+1)*(Product_{i=0..n} (n+i)!)/Product_{i=1..n+1} i!.
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) ~ 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
|
|
|
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
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
|
|
CROSSREFS
|
Central column in triangle A009963.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|