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A006366
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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)
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6
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1, 2, 5, 20, 132, 1452, 26741, 826540, 42939620, 3752922788, 552176360205, 136830327773400, 57125602787130000, 40191587143536420000, 47663133295107416936400, 95288872904963020131203520, 321195665986577042490185260608
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OFFSET
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0,2
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COMMENTS
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In the 1995 Encyclopedia of Integer Sequences this sequence appears twice, as both M1529 and M1530.
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REFERENCES
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G. E. Andrews, Plane partitions (III): the Weak Macdonald Conjecture, Invent. Math., 53 (1979), 193-225.
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.
W. F. Lunnon "The Pascal matrix", Fib. Quart. vol. 15 (1977) pp. 201-204.
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
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Vincenzo Librandi, Table of n, a(n) for n = 0..90
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv math.CO/0008184
P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, Determinant formulae for some tiling problems...
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FORMULA
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Product( (3*i-1)/(3*i-2) * Product ( (n+i+j-1)/(2*i+j-1), j=i..n), i=1..n).
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
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=prod(i=0, n-1, (3*i+2)*(3*i)!/(n+i)!)
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CROSSREFS
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Cf. A005130, also A003827, A005156, A005158, A005160-A005164, A048601, A050204.
Sequence in context: A076795 A130293 A156073 * A012317 A118181 A140988
Adjacent sequences: A006363 A006364 A006365 * A006367 A006368 A006369
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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