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A051255 Number of cyclically symmetric transpose complement plane partitions in a 2n X 2n X 2n box. 9
1, 1, 2, 11, 170, 7429, 920460, 323801820, 323674802088, 919856004546820, 7434724817843114428, 170943292930264547814443, 11183057455425265737399150652, 2081853548182272792243789109645876 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.15), p. 199 (corrected).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..60

M. T. Batchelor, J. de Gier and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv cond-mat/0101385 (see N_8(2n))

D. M. Bressoud, Corrections: Proofs and Confirmations

N. T. Cameron, Random walks, trees and extensions of Riordan group techniques

J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math.CO/0211285, 2002.

I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.

FORMULA

a(n) ~ exp(1/72) * GAMMA(1/3)^(2/3) * n^(7/72) * 3^(3*n^2 - 3*n/2 + 11/72) / (A^(1/6) * Pi^(1/3) * 2^(4*n^2 - n - 1/18)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 28 2015

EXAMPLE

For n=0 there is the empty partition by convention so a(0)=1. For n=1 there is a single cyclically symmetric transpose complement plane partition in a 2 X 2 X 2 box so a(1)=1.

MAPLE

A051255 := proc(n) local i; mul((3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!), i=0..n-1); end;

MATHEMATICA

a[n_] := Product[(3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!), {i, 0, n-1}]; Table[a[n], {n, 0, 13}] (* Jean-Fran├žois Alcover, Feb 25 2014 *)

PROG

(PARI) a(n)=prod(i=0, n-1, (3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!)); \\ Joerg Arndt, Feb 25 2014

CROSSREFS

Cf. A049504.

Sequence in context: A067968 A197336 A013050 * A120445 A003088 A121231

Adjacent sequences:  A051252 A051253 A051254 * A051256 A051257 A051258

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org)

Added missing a(0)=1 term. - Michael Somos, Feb 25 2014

STATUS

approved

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Last modified April 29 19:06 EDT 2017. Contains 285613 sequences.