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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).
(Formerly M3115)
13

%I M3115

%S 1,1,3,26,646,45885,9304650,5382618660,8878734657276,

%T 41748486581283118,559463042542694360707,21363742267675013243931852,

%U 2324392978926652820310084179576,720494439459132215692530771292602232,636225819409712640497085074811372777428304

%N 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).

%C a(n+1) is the Hankel transform of A006013. - _Paul Barry_, Jan 20 2007

%C a(n+1) is the Hankel transform of A025174(n+1). - _Paul Barry_, Apr 14 2008

%D Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 201, VS(2n+1).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D 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 Gheorghe Coserea, <a href="/A005156/b005156.txt">Table of n, a(n) for n = 0..66</a>

%H M. T. Batchelor, J. de Gier and B. Nienhuis, <a href="http://arXiv.org/abs/cond-mat/0101385">The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions</a>, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001, (see A_V(2n+1)).

%H N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">Random walks, trees and extensions of Riordan group techniques</a>

%H J. de Gier, <a href="http://arXiv.org/abs/math.CO/0211285">Loops, matchings and alternating-sign matrices</a>, arXiv:math/0211285 [math.CO], 2002-2003.

%H I. Fischer, <a href="http://arXiv.org/abs/math.CO/0501102">The number of monotone triangles with prescribed bottom row</a>, arXiv:math/0501102 [math.CO], 2005.

%H I. Gessel and G. Xin, <a href="http://arXiv.org/abs/math.CO/0505217">The generating function of ternary trees and continued fractions</a>, arXiv:math/0505217 [math.CO], 2005.

%H W. Hebsich and M. Rubey, <a href="http://arxiv.org/abs/math/0702086">Extended Rate, More Gfun</a>, arXiv:math/0702086 [math.CO], 2007. [See p. 23.]

%H G. Kuperberg, <a href="http://arXiv.org/abs/math.CO/0008184">Symmetry classes of alternating-sign matrices under one roof</a>, arXiv:math/0008184 [math.CO], 2000-2001, (see A_V(2n+1)).

%H A. V. Razumov and Yu. G. Stroganov, <a href="http://arXiv.org/abs/math-ph/0312071">On refined enumerations of some symmetry classes of alternating sign matrices</a>, arXiv:math-ph/0312071, 2003.

%H D. P. Robbins, <a href="http://arXiv.org/abs/math.CO/0008045">Symmetry classes of alternating sign matrices</a>, arXiv:math/0008045 [math.CO], 2000.

%H R. P. Stanley, <a href="/A005130/a005130.pdf">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

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

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

%F 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

%p 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;

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

%o (PARI) a(n) = prod(k = 0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!));

%o vector(15, n, a(n-1)) \\ _Gheorghe Coserea_, May 30 2016

%Y Cf. A109074, A134357.

%K nonn,nice,easy

%O 0,3

%A _N. J. A. Sloane_

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