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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

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

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

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 [cond-mat.stat-mech], 2001. See N_8(2n).

D. M. Bressoud, Corrections: Proofs and Confirmations

N. T. Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.

Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.

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

a(n) = Product_{i=0..n-1} (3i+1) C(6i,2i)/(C(4i+1,2i)*(2i+1)), using [Bressoud, Corrections, p. 199: N8]. - M. F. Hasler, Oct 04 2018

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

(PARI) A051255(n)=prod(i=0, n-1, (3*i+1)*binomial(6*i, 2*i)/binomial(4*i+1, 2*i)/(2*i+1)) \\ M. F. Hasler, Oct 04 2018

CROSSREFS

Cf. A049504.

Sequence in context: A295269 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

Missing a(0)=1 term added by Michael Somos, Feb 25 2014

STATUS

approved

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Last modified June 16 10:03 EDT 2019. Contains 324152 sequences. (Running on oeis4.)