A059475 Number of 2n X 2n half-turn symmetric alternating-sign matrices (HTSASM's).

1, 2, 10, 140, 5544, 622908, 198846076, 180473355920, 465904151957920, 3422048076740462480, 71525763221287897903500, 4254840960508487045451825000, 720428791920558617462950575000000, 347230535542092373572967034254050000000

Offset: 0

Author

N. J. A. Sloane, Feb 04 2001

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..66
• Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
• J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
• Heuer, Dylan, Chelsey Morrow, Ben Noteboom, Sara Solhjem, Jessica Striker, and Corey Vorland. "Chained permutations and alternating sign matrices - Inspired by three-person chess." Discrete Mathematics 340, no. 12 (2017): 2732-2752. Also arXiv:1611.03387.
• G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2001.

Crossrefs

Even-numbered terms of A005158.

Cf. A005130, A006366, A049503.

Programs

Mathematica

a[n_] := Product[(3k+1)(3k+2)(3k)!^2/(n+k)!^2, {k, 0, n-1}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Sep 01 2018, after Seiichi Manyama *)

Formula

a(n) = A005130(n)*A006366(n).

a(n) = A049503(n)*Product_{k=0..n-1} (3*k+2)/(3*k+1). - Seiichi Manyama, Jul 29 2018

a(n) ~ exp(1/18) * Gamma(1/3)^(2/3) * n^(1/18) * 3^(3*n^2 + 1/9) / (A^(2/3) * Pi^(1/3) * 2^(4*n^2 + 1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 26 2020