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

%I A005161 M1700 #47 Feb 19 2025 12:06:18
%S A005161 1,1,1,2,6,33,286,4420,109820,4799134,340879665,42235307100,
%T A005161 8564558139000,3012862604463000,1742901718473961200,
%U A005161 1742218029490675762080,2873822682985675809192288,8167157387273280570395662320,38402596062535617548517706584760,310388509293255836481583597538626504
%N A005161 Number of alternating sign 2n+1 X 2n+1 matrices symmetric with respect to both horizontal and vertical axes (VHSASM's).
%D A005161 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005161 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.
%H A005161 Andrew Howroyd, <a href="/A005161/b005161.txt">Table of n, a(n) for n = 0..80</a>
%H A005161 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H A005161 Paul Barry, <a href="https://doi.org/10.3390/math13020242">Extensions of Riordan Arrays and Their Applications</a>, Mathematics (2025) Vol. 13, No. 2, 242. See p. 24.
%H A005161 Ira Gessel and Guoce Xin, <a href="https://arxiv.org/abs/math/0505217">The generating function of ternary trees and continued fractions</a>, arXiv:math/0505217 [math.CO], 2005.
%H A005161 Soichi Okada, <a href="https://doi.org/10.1007/s10801-006-6028-3">Enumeration of symmetry classes of alternating sign matrices and characters of classical groups</a>, Journal of Algebraic Combinatorics volume 23, pages 43-69 (2006).
%H A005161 Pavel Pyatov, <a href="https://arxiv.org/abs/math-ph/0406025">Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon</a>, arXiv:math-ph/0406025, 2004. [_Vladeta Jovovic_, Aug 15 2008]
%H A005161 David P. Robbins, <a href="https://arxiv.org/abs/math/0008045">Symmetry classes of alternating sign matrices</a>, arXiv:math/0008045 [math.CO], 2000.
%H A005161 Richard P. Stanley, <a href="/A005130/a005130.pdf">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire énumérative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]
%F A005161 Robbins gives a simple (conjectured) formula, which was proven by Okada.
%F A005161 a(2*n) = A005156(n)*A051255(n); a(2*n+1) = A005156(n)*A051255(n+1). - _Paul Zinn-Justin_, May 05 2023
%F A005161 a(n) = A005156(floor(n/2)) * A051255(ceiling(n/2)). - _Andrew Howroyd_, May 09 2023
%o A005161 (PARI) \\ here b(n) and c(n) are A005156 and A051255.
%o A005161 b(n) = prod(k=0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!));
%o A005161 c(n) = prod(k=0, n-1, (3*k+1)*(6*k)!*(2*k)!/((4*k)!*(4*k+1)!));
%o A005161 a(n) = b(n\2) * c((n+1)\2) \\ _Andrew Howroyd_, May 09 2023
%Y A005161 Cf. A005130, A005162, A005163, A005156, A051255, A059476.
%K A005161 nonn,nice
%O A005161 0,4
%A A005161 _N. J. A. Sloane_
%E A005161 More terms (from the P. Pyatov paper) from _Vladeta Jovovic_, Aug 15 2008
%E A005161 Terms a(13) and beyond from _Andrew Howroyd_, May 09 2023