%I #13 Mar 05 2021 07:05:51

%S 1,1,2,14,588,252252,1875745872,409565359659456,

%T 4443872618422784042496,4052080633200943761869999708160,

%U 524883317743439723147432404145717855232000,16321637725818077271987866314412476606229589461376000000

%N Product of first n Robbins numbers.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlternatingSignMatrix.html">Alternating Sign Matrix</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>.

%F a(n) = Product_{k=1..n} A005130(k).

%F 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.

%p b:= proc(n) option remember; `if`(n<2, 1, b(n-1)*

%p (n-1)!*(3*n-2)!/((2*n-2)!*(2*n-1)!))

%p end:

%p a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:

%p seq(a(n), n=0..12); # _Alois P. Heinz_, Mar 04 2021

%t Table[Product[Product[(3*j + 1)!/(k + j)!, {j, 0, k-1}], {k,1,n}], {n,0,12}]

%t FoldList[Times, 1, Table[Product[(3*j + 1)!/(n + j)!, {j, 0, n - 1}], {n, 1, 12}]]

%Y Cf. A005130.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 04 2021