A005161 Number of alternating sign 2n+1 X 2n+1 matrices symmetric with respect to both horizontal and vertical axes (VHSASM's).
1, 1, 1, 2, 6, 33, 286, 4420, 109820, 4799134, 340879665, 42235307100, 8564558139000, 3012862604463000, 1742901718473961200, 1742218029490675762080, 2873822682985675809192288, 8167157387273280570395662320, 38402596062535617548517706584760, 310388509293255836481583597538626504
Offset: 0
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- 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.
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..80
- 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, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 24.
- Ira Gessel and Guoce Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
- Soichi Okada, Enumeration of symmetry classes of alternating sign matrices and characters of classical groups, Journal of Algebraic Combinatorics volume 23, pages 43-69 (2006).
- Pavel Pyatov, Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon, arXiv:math-ph/0406025, 2004. [_Vladeta Jovovic_, Aug 15 2008]
- David P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
- Richard P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire énumérative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]
Programs
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PARI
\\ here b(n) and c(n) are A005156 and A051255. b(n) = prod(k=0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!)); c(n) = prod(k=0, n-1, (3*k+1)*(6*k)!*(2*k)!/((4*k)!*(4*k+1)!)); a(n) = b(n\2) * c((n+1)\2) \\ Andrew Howroyd, May 09 2023
Formula
Robbins gives a simple (conjectured) formula, which was proven by Okada.
Extensions
More terms (from the P. Pyatov paper) from Vladeta Jovovic, Aug 15 2008
Terms a(13) and beyond from Andrew Howroyd, May 09 2023