%I A005156 M3115
%S A005156 1,1,3,26,646,45885,9304650,5382618660,8878734657276,41748486581283118,
%T A005156 559463042542694360707,21363742267675013243931852,2324392978926652820310084179576,
%U A005156 720494439459132215692530771292602232,636225819409712640497085074811372777428304
%N 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).
%C A005156 a(n+1) is the Hankel transform of A006013. - Paul Barry (pbarry(AT)wit.ie), 
               Jan 20 2007
%C A005156 a(n+1) is the Hankel transform of A025174(n+1). - Paul Barry (pbarry(AT)wit.ie), 
               Apr 14 2008
%D A005156 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005156 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 
               201, VS(2n+1).
%D A005156 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 A005156 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, arXiv cond-mat/
               0101385</a> (see A_V(2n+1)).
%H A005156 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">
               Random walks, trees and extensions of Riordan group techniques</a>
%H A005156 I. Fischer, <a href="http://arXiv.org/abs/math.CO/0501102">The number 
               of monotone triangles with prescribed bottom row</a>
%H A005156 I. Gessel and G. Xin, <a href="http://arXiv.org/abs/math.CO/0505217">
               The generating function of ternary trees and continued fractions</
               a>
%H A005156 J. de Gier, <a href="http://arXiv.org/abs/math.CO/0211285">Loops, matchings 
               and alternating-sign matrices</a>
%H A005156 G. Kuperberg, <a href="http://arXiv.org/abs/math.CO/0008184">Symmetry 
               classes of alternating-sign matrices under one roof, arXiv math.CO/
               0008184</a> (see A_V(2n+1)).
%H A005156 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>
%H A005156 D. P. Robbins, Symmetry classes of alternating sign matrices, <a href="http:/
               /arXiv.org/abs/math.CO/0008045">arXiv:math.CO/0008045</a>
%F A005156 The formula for a(n) (see the Maple code) was conjectured by Robbins 
               and proved by Kuperberg.
%F A005156 (1/2^n) * prod[k=1..n, {(6k-2)!(2k-1)!}/{(4k-1)!(4k-2)!}] (Razumov/Stroganov).
%p A005156 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;
%Y A005156 Cf. A109074/A134357.
%Y A005156 Sequence in context: A064941 A112612 A129430 this_sequence A101613 A088730 
               A058956
%Y A005156 Adjacent sequences: A005153 A005154 A005155 this_sequence A005157 A005158 
               A005159
%K A005156 nonn,nice,easy
%O A005156 0,3
%A A005156 N. J. A. Sloane (njas(AT)research.att.com).