A005156 Number of alternating sign 2n+1 X 2n+1 matrices symmetric about the vertical axis (VSASM's); also 2n X 2n off-diagonally symmetric alternating sign matrices (OSASM's).

1, 1, 3, 26, 646, 45885, 9304650, 5382618660, 8878734657276, 41748486581283118, 559463042542694360707, 21363742267675013243931852, 2324392978926652820310084179576, 720494439459132215692530771292602232, 636225819409712640497085074811372777428304

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

a(n+1) is the Hankel transform of A006013. - Paul Barry, Jan 20 2007

a(n+1) is the Hankel transform of A025174(n+1). - Paul Barry, Apr 14 2008

References

• D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 201, VS(2n+1).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Gheorghe Coserea, 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.
• Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 23.
• M. T. Batchelor, J. de Gier and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001, (see A_V(2n+1)).
• N. T. Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.
• J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
• I. Fischer, The number of monotone triangles with prescribed bottom row, arXiv:math/0501102 [math.CO], 2005.
• I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
• W. Hebsich and M. Rubey, Extended Rate, More Gfun, arXiv:math/0702086 [math.CO], 2007. [See p. 23.]
• G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001, (see A_V(2n+1)).
• A. V. Razumov and Yu. G. Stroganov, On refined enumerations of some symmetry classes of alternating sign matrices, arXiv:math-ph/0312071, 2003.
• D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
• R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
• R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

Crossrefs

Cf. A109074, A134357.

Programs

Maple

A005156 := proc(n) local i,j,t1; (-3)^(n^2)*mul( mul( (6*j-3*i+1)/(2*j-i+2*n+1), j=1..n ),i=1..2*n+1); end;

Mathematica

Table[1/2^n Product[((6k-2)!(2k-1)!)/((4k-1)!(4k-2)!),{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 07 2011 *)

PARI

a(n) = prod(k = 0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!));
vector(15, n, a(n-1))  \\ Gheorghe Coserea, May 30 2016

Formula

The formula for a(n) (see the Maple code) was conjectured by Robbins and proved by Kuperberg.

a(n) = (1/2^n) * Product_{k=1..n} ((6k-2)!(2k-1)!)/((4k-1)!(4k-2)!) (Razumov/Stroganov).

a(n) ~ exp(1/72) * Pi^(1/6) * 3^(3*n^2 + 3*n/2 + 11/72) / (A^(1/6) * GAMMA(1/3)^(1/3) * n^(5/72) * 2^(4*n^2 + 3*n + 1/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015