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A006366 Number of cyclically symmetric plane partitions in the n-cube; also number of 2n X 2n half-turn symmetric alternating sign matrices divided by number of n X n alternating sign matrices.
(Formerly M1529)
6
1, 2, 5, 20, 132, 1452, 26741, 826540, 42939620, 3752922788, 552176360205, 136830327773400, 57125602787130000, 40191587143536420000, 47663133295107416936400, 95288872904963020131203520, 321195665986577042490185260608 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
In the 1995 Encyclopedia of Integer Sequences this sequence appears twice, as both M1529 and M1530.
REFERENCES
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.7) on page 198, except the formula given is incorrect. It should be as shown here.
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
G. E. Andrews,Plane partitions (III): the Weak Macdonald Conjecture, Invent. Math., 53 (1979), 193-225.
P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops, arXiv:math-ph/0410002, 2004.
Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001.
W. F. Lunnon, The Pascal matrix, Fib. Quart. vol. 15 (1977) pp. 201-204.
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]
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
FORMULA
a(n) = Product_{i=1..n} (((3*i-1)/(3*i-2))*Product_{j=i..n} (n+i+j-1)/(2*i+j-1)).
a(n) ~ exp(1/36) * GAMMA(1/3)^(4/3) * n^(7/36) * 3^(3*n^2/2 + 11/36) / (A^(1/3) * Pi^(2/3) * 2^(2*n^2 + 7/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015
MAPLE
A006366 := proc(n) local i, j; mul((3*i - 1)*mul((n + i + j - 1)/(2*i + j - 1), j = i .. n)/(3*i - 2), i = 1 .. n) end;
MATHEMATICA
Table[Product[(3i-1)/(3i-2) Product[(n+i+j-1)/(2i+j-1), {j, i, n}], {i, n}], {n, 0, 20}] (* Harvey P. Dale, Apr 17 2013 *)
PROG
(PARI) a(n)=prod(i=0, n-1, (3*i+2)*(3*i)!/(n+i)!)
CROSSREFS
Sequence in context: A076795 A130293 A156073 * A012317 A297630 A297629
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)