|
Search: id:A005156
|
|
|
| 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)
|
|
+0
12
|
|
| 1, 1, 3, 26, 646, 45885, 9304650, 5382618660, 8878734657276, 41748486581283118, 559463042542694360707, 21363742267675013243931852, 2324392978926652820310084179576, 720494439459132215692530771292602232, 636225819409712640497085074811372777428304
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
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
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 201, VS(2n+1).
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
|
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
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
J. de Gier, Loops, matchings and alternating-sign matrices
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
|
|
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).
|
|
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;
|
|
CROSSREFS
|
Cf. A109074/A134357.
Sequence in context: A064941 A112612 A129430 this_sequence A101613 A088730 A058956
Adjacent sequences: A005153 A005154 A005155 this_sequence A005157 A005158 A005159
|
|
KEYWORD
|
nonn,nice,easy
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
Search completed in 0.002 seconds
|
