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A342181 Product of first n Robbins numbers.

%I A342181 #14 Feb 16 2025 08:34:01
%S A342181 1,1,2,14,588,252252,1875745872,409565359659456,
%T A342181 4443872618422784042496,4052080633200943761869999708160,
%U A342181 524883317743439723147432404145717855232000,16321637725818077271987866314412476606229589461376000000
%N A342181 Product of first n Robbins numbers.
%H A342181 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingSignMatrix.html">Alternating Sign Matrix</a>.
%H A342181 Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>.
%F A342181 a(n) = Product_{k=1..n} A005130(k).
%F A342181 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 A342181 b:= proc(n) option remember; `if`(n<2, 1, b(n-1)*
%p A342181       (n-1)!*(3*n-2)!/((2*n-2)!*(2*n-1)!))
%p A342181     end:
%p A342181 a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:
%p A342181 seq(a(n), n=0..12);  # _Alois P. Heinz_, Mar 04 2021
%t A342181 Table[Product[Product[(3*j + 1)!/(k + j)!, {j, 0, k-1}], {k,1,n}], {n,0,12}]
%t A342181 FoldList[Times, 1, Table[Product[(3*j + 1)!/(n + j)!, {j, 0, n - 1}], {n, 1, 12}]]
%Y A342181 Cf. A005130.
%K A342181 nonn
%O A342181 0,3
%A A342181 _Vaclav Kotesovec_, Mar 04 2021