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A266091 a(n) = Product_{k=0..n} (3*k)!/(n+k)!. 1
1, 3, 15, 126, 1782, 42471, 1706562, 115640460, 13216815036, 2548124192970, 828751754742975, 454739496669274500, 420972227408592675000, 657522745057190417409000, 1732789066323343611643088400, 7704900186426840030325195822560, 57807195523790513335568376591463776 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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.
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
MATHEMATICA
Table[Product[(3 k)!/(n + k)!, {k, 0, n}], {n, 0, 16}] (* Vincenzo Librandi, Dec 21 2015 *)
PROG
(PARI) a(n) = prod(k=0, n, (3*k)!/(n+k)!);
(Magma) [&*[Factorial(3*k)/Factorial(n+k): k in [0..n]]: n in [0..16]]; // Vincenzo Librandi, Dec 21 2015
CROSSREFS
Sequence in context: A270503 A212553 A229673 * A135255 A182489 A330804
KEYWORD
nonn,easy
AUTHOR
Michel Marcus, Dec 21 2015
STATUS
approved

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Last modified July 17 18:09 EDT 2024. Contains 374377 sequences. (Running on oeis4.)