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a(n) = Product_{k=0..n} (3*k)!/(n+k)!.
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%I #28 Feb 21 2023 11:33:08

%S 1,3,15,126,1782,42471,1706562,115640460,13216815036,2548124192970,

%T 828751754742975,454739496669274500,420972227408592675000,

%U 657522745057190417409000,1732789066323343611643088400,7704900186426840030325195822560,57807195523790513335568376591463776

%N a(n) = Product_{k=0..n} (3*k)!/(n+k)!.

%C a(n) gives the number of diagonally and antidiagonally symmetric alternating sign matrices (DASASM's) of order (2n+1) X (2n+1) (see Behrend et al. link).

%H Roger E. Behrend, Ilse Fischer, Matjaž Konvalinka, <a href="http://arxiv.org/abs/1512.06030">Diagonally and antidiagonally symmetric alternating sign matrices of odd order</a>, arXiv:1512.06030 [math.CO], 2015.

%F a(n) ~ Gamma(1/3)^(1/3) * exp(1/36) * n^(1/36) * 3^(3*n^2/2 + 2*n + 11/36) / (A^(1/3) * Pi^(1/6) * 2^(2*n^2 + 2*n + 7/12)), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Dec 21 2015

%F a(n) = Product_{1 <= i <= j <= n} (i + 2*j)/(i + j - 1). Note that Product_{1 <= i <= j <= n} (i + j)/(i + j - 1) = 2^n. - _Peter Bala_, Feb 19 2023

%t Table[Product[(3 k)!/(n + k)!, {k, 0, n}], {n, 0, 16}] (* _Vincenzo Librandi_, Dec 21 2015 *)

%o (PARI) a(n) = prod(k=0, n, (3*k)!/(n+k)!);

%o (Magma) [&*[Factorial(3*k)/Factorial(n+k): k in [0..n]]: n in [0..16]]; // _Vincenzo Librandi_, Dec 21 2015

%Y Cf. A005157, A086205.

%K nonn,easy

%O 0,2

%A _Michel Marcus_, Dec 21 2015