A051255 Number of cyclically symmetric transpose complement plane partitions in a 2n X 2n X 2n box.
1, 1, 2, 11, 170, 7429, 920460, 323801820, 323674802088, 919856004546820, 7434724817843114428, 170943292930264547814443, 11183057455425265737399150652, 2081853548182272792243789109645876
Offset: 0
Examples
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. G.f. = 1 + x + 2*x^2 + 11*x^3 + 170*x^4 + 7429*x^5 + 920460*x^6 + 323801820*x^7 + ... - _Michael Somos_, May 15 2022
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.
- Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
- Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 24.
- 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 and 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.
- Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
- Yaping Liu, On the Recursiveness of Pascal Sequences, Global J. of Pure and Appl. Math. (2022) Vol. 18, No. 1, 71-80.
Programs
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Maple
A051255 := proc(n) local i; mul((3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!),i=0..n-1); end;
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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-François Alcover, Feb 25 2014 *) -
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
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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
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
Extensions
More terms from Michel ten Voorde
Missing a(0)=1 term added by Michael Somos, Feb 25 2014
Comments