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A059476 -id:A059476 - OEIS
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Search: a059476 -id:a059476
Displaying 1-2 of 2 results found. page 1
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A005160 Number of alternating sign n X n matrices invariant under a quarter turn.
(Formerly M0396) 		+10
4
1, 0, 1, 2, 3, 0, 12, 40, 100, 0, 1225, 6860, 28812, 0, 1037232, 9779616 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Robbins incorrectly gives a(12) = 6460.
REFERENCES
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.
LINKS
Table of n, a(n) for n=1..16.
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001.
D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
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. Preprint. [Annotated scanned copy]
FORMULA
Robbins gives a simple (conjectured) formula.
CROSSREFS
a(4n) gives A059476.
KEYWORD
nonn,nice,more
AUTHOR
N. J. A. Sloane
STATUS
approved
A005161 Number of alternating sign 2n+1 X 2n+1 matrices symmetric with respect to both horizontal and vertical axes (VHSASM's).
(Formerly M1700) 		+10
3
1, 1, 1, 2, 6, 33, 286, 4420, 109820, 4799134, 340879665, 42235307100, 8564558139000, 3012862604463000, 1742901718473961200, 1742218029490675762080, 2873822682985675809192288, 8167157387273280570395662320, 38402596062535617548517706584760, 310388509293255836481583597538626504 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
REFERENCES
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..80
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
Soichi Okada, Enumeration of symmetry classes of alternating sign matrices and characters of classical groups, Journal of Algebraic Combinatorics volume 23, pages 43-69 (2006).
P. Pyatov, Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon, arXiv:math-ph/0406025, 2004. [Vladeta Jovovic, Aug 15 2008]
D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
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. Preprint. [Annotated scanned copy]
FORMULA
Robbins gives a simple (conjectured) formula, which was proven by Okada.
a(2*n) = A005156(n)*A051255(n); a(2*n+1) = A005156(n)*A051255(n+1). - Paul Zinn-Justin, May 05 2023
a(n) = A005156(floor(n/2)) * A051255(ceiling(n/2)). - Andrew Howroyd, May 09 2023
PROG
(PARI) \\ here b(n) and c(n) are A005156 and A051255.
b(n) = prod(k=0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!));
c(n) = prod(k=0, n-1, (3*k+1)*(6*k)!*(2*k)!/((4*k)!*(4*k+1)!));
a(n) = b(n\2) * c((n+1)\2) \\ Andrew Howroyd, May 09 2023
CROSSREFS
Cf. A005130, A005162, A005163, A005156, A051255, A059476.
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms (from the P. Pyatov paper) from Vladeta Jovovic, Aug 15 2008
Terms a(13) and beyond from Andrew Howroyd, May 09 2023
STATUS
approved
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Last modified September 7 11:06 EDT 2024. Contains 375730 sequences. (Running on oeis4.)