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%I M1529 #38 Aug 22 2022 07:02:20
%S 1,2,5,20,132,1452,26741,826540,42939620,3752922788,552176360205,
%T 136830327773400,57125602787130000,40191587143536420000,
%U 47663133295107416936400,95288872904963020131203520,321195665986577042490185260608
%N 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.
%C In the 1995 Encyclopedia of Integer Sequences this sequence appears twice, as both M1529 and M1530.
%D 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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.
%H Vincenzo Librandi, <a href="/A006366/b006366.txt">Table of n, a(n) for n = 0..90</a>
%H G. E. Andrews,<a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002095335">Plane partitions (III): the Weak Macdonald Conjecture</a>, Invent. Math., 53 (1979), 193-225.
%H P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, <a href="https://arxiv.org/abs/math-ph/0410002">Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops</a>, arXiv:math-ph/0410002, 2004.
%H Anatol N. Kirillov, <a href="https://doi.org/10.3842/SIGMA.2016.034">Notes on Schubert, Grothendieck and key polynomials</a>, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
%H G. Kuperberg, <a href="https://arxiv.org/abs/math/0008184">Symmetry classes of alternating-sign matrices under one roof</a>, arXiv:math/0008184 [math.CO], 2000-2001.
%H W. F. Lunnon, <a href="http://www.fq.math.ca/Scanned/15-3/lunnon.pdf">The Pascal matrix</a>, Fib. Quart. vol. 15 (1977) pp. 201-204.
%H R. P. Stanley, <a href="/A005130/a005130.pdf">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]
%H P. J. Taylor, <a href="http://cheddarmonk.org/papers/distinct-dimer-hex-tilings.pdf">Counting distinct dimer hex tilings</a>, Preprint, 2015.
%F a(n) = Product_{i=1..n} (((3*i-1)/(3*i-2))*Product_{j=i..n} (n+i+j-1)/(2*i+j-1)).
%F 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
%p 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;
%t 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 *)
%o (PARI) a(n)=prod(i=0,n-1,(3*i+2)*(3*i)!/(n+i)!)
%Y Cf. A005130, also A003827, A005156, A005158, A005160-A005164, A048601, A050204.
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_
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