A005161 Number of alternating sign 2n+1 X 2n+1 matrices symmetric with respect to both horizontal and vertical axes (VHSASM's). (Formerly M1700)

1, 1, 1, 2, 6, 33, 286, 4420, 109820, 4799134, 340879665, 42235307100, 8564558139000, 3012862604463000, 1742901718473961200, 1742218029490675762080, 2873822682985675809192288, 8167157387273280570395662320, 38402596062535617548517706584760, 310388509293255836481583597538626504

Offset

0,4

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.

I. Gessel and G. 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).

P. Pyatov, Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon, arXiv:math-ph/0406025, 2004. [Vladeta Jovovic, Aug 15 2008]

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. Preprint. [Annotated scanned copy]

Formula

Robbins gives a simple (conjectured) formula, which was proven by Okada.

a(2*n) = A005156(n)*A051255(n); a(2*n+1) = A005156(n)*A051255(n+1). - Paul Zinn-Justin, May 05 2023

a(n) = A005156(floor(n/2)) * A051255(ceiling(n/2)). - Andrew Howroyd, May 09 2023

Prog

(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

Crossrefs

Cf. A005130, A005162, A005163, A005156, A051255, A059476.

Sequence in context: A053042 A174432 A012874 * A062970 A259436 A278611

Adjacent sequences: A005158 A005159 A005160 * A005162 A005163 A005164

Keyword

nonn,nice

Author

N. J. A. Sloane

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

Status

approved