A342181 Product of first n Robbins numbers.

1, 1, 2, 14, 588, 252252, 1875745872, 409565359659456, 4443872618422784042496, 4052080633200943761869999708160, 524883317743439723147432404145717855232000, 16321637725818077271987866314412476606229589461376000000

Offset: 0

Author

Vaclav Kotesovec, Mar 04 2021

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Alternating Sign Matrix.
• Wikipedia, Barnes G-function.

Crossrefs

Cf. A005130.

Programs

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}]]

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.