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)

1, 2, 5, 20, 132, 1452, 26741, 826540, 42939620, 3752922788, 552176360205, 136830327773400, 57125602787130000, 40191587143536420000, 47663133295107416936400, 95288872904963020131203520, 321195665986577042490185260608

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

Vincenzo Librandi, Table of n, a(n) for n = 0..90

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

Cf. A005130, also A003827, A005156, A005158, A005160-A005164, A048601, A050204.

Sequence in context: A076795 A130293 A156073 * A012317 A297630 A297629

Adjacent sequences: A006363 A006364 A006365 * A006367 A006368 A006369

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved