A266091 a(n) = Product_{k=0..n} (3*k)!/(n+k)!.

1, 3, 15, 126, 1782, 42471, 1706562, 115640460, 13216815036, 2548124192970, 828751754742975, 454739496669274500, 420972227408592675000, 657522745057190417409000, 1732789066323343611643088400, 7704900186426840030325195822560, 57807195523790513335568376591463776

Offset: 0

Author

Michel Marcus, Dec 21 2015

Keywords

• nonn
• easy

Comments

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).

Links

• Roger E. Behrend, Ilse Fischer, Matjaž Konvalinka, Diagonally and antidiagonally symmetric alternating sign matrices of odd order, arXiv:1512.06030 [math.CO], 2015.

Crossrefs

Cf. A005157, A086205.

Programs

Magma

[&*[Factorial(3*k)/Factorial(n+k): k in [0..n]]: n in [0..16]]; // Vincenzo Librandi, Dec 21 2015

Mathematica

Table[Product[(3 k)!/(n + k)!, {k, 0, n}], {n, 0, 16}] (* Vincenzo Librandi, Dec 21 2015 *)

PARI

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

Formula

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

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