OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Alternating Sign Matrix.
Wikipedia, Barnes G-function.
FORMULA
a(n) = Product_{k=1..n} A005130(k).
a(n) ~ Pi^(n/3 + 1/6) * 3^(n^3/2 + 3*n^2/4 + n/18 - 13/216) * exp(n/6 + 11*zeta(3)/(144*Pi^2) + 19/216) / (BarnesG(1/3)^(2/3) * n^(5*n/36 + 5/72) * 2^(2*n^3/3 + n^2 - n/12 - 1/12) * A^(n/3 + 19/18) * Gamma(1/3)^(2*n/3 + 7/9)), where A is the Glaisher-Kinkelin constant A074962.
MAPLE
b:= proc(n) option remember; `if`(n<2, 1, b(n-1)*
(n-1)!*(3*n-2)!/((2*n-2)!*(2*n-1)!))
end:
a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:
seq(a(n), n=0..12); # Alois P. Heinz, Mar 04 2021
MATHEMATICA
Table[Product[Product[(3*j + 1)!/(k + j)!, {j, 0, k-1}], {k, 1, n}], {n, 0, 12}]
FoldList[Times, 1, Table[Product[(3*j + 1)!/(n + j)!, {j, 0, n - 1}], {n, 1, 12}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 04 2021
STATUS
approved