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A005156
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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)
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13
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1, 1, 3, 26, 646, 45885, 9304650, 5382618660, 8878734657276, 41748486581283118, 559463042542694360707, 21363742267675013243931852, 2324392978926652820310084179576, 720494439459132215692530771292602232, 636225819409712640497085074811372777428304
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n+1) is the Hankel transform of A006013. - Paul Barry (pbarry(AT)wit.ie), Jan 20 2007
a(n+1) is the Hankel transform of A025174(n+1). - Paul Barry (pbarry(AT)wit.ie), Apr 14 2008
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REFERENCES
| D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 201, VS(2n+1).
W. Hebsich and M. Rubey, Extended Rate, More Gfun, http://arxiv.org/PS_cache/math/pdf/0702/0702086v2.pdf. [See p. 23.]
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.
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LINKS
| M. T. Batchelor, J. de Gier and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv cond-mat/0101385 (see A_V(2n+1)).
N. T. Cameron, Random walks, trees and extensions of Riordan group techniques
J. de Gier, Loops, matchings and alternating-sign matrices
I. Fischer, The number of monotone triangles with prescribed bottom row
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv math.CO/0008184 (see A_V(2n+1)).
A. V. Razumov and Yu. G. Stroganov, On refined enumerations of some symmetry classes of alternating sign matrices
D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math.CO/0008045
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FORMULA
| The formula for a(n) (see the Maple code) was conjectured by Robbins and proved by Kuperberg.
(1/2^n) * prod[k=1..n, {(6k-2)!(2k-1)!}/{(4k-1)!(4k-2)!}] (Razumov/Stroganov).
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MAPLE
| 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;
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MATHEMATICA
| Table[1/2^n Product[((6k-2)!(2k-1)!)/((4k-1)!(4k-2)!), {k, n}], {n, 0, 20}] (* From Harvey P. Dale, Jul 07 2011 *)
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CROSSREFS
| Cf. A109074/A134357.
Sequence in context: A064941 A112612 A129430 * A101613 A174811 A088730
Adjacent sequences: A005153 A005154 A005155 * A005157 A005158 A005159
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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