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%I #69 Aug 22 2022 07:02:57
%S 1,1,2,11,170,7429,920460,323801820,323674802088,919856004546820,
%T 7434724817843114428,170943292930264547814443,
%U 11183057455425265737399150652,2081853548182272792243789109645876
%N Number of cyclically symmetric transpose complement plane partitions in a 2n X 2n X 2n box.
%C Hankel transform of A006013 without initial term is this sequence without initial term. - _Michael Somos_, May 15 2022
%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.15), p. 199 (corrected).
%H Vincenzo Librandi, <a href="/A051255/b051255.txt">Table of n, a(n) for n = 0..60</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.
%H M. T. Batchelor, J. de Gier and B. Nienhuis, <a href="http://arXiv.org/abs/cond-mat/0101385">The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions</a>, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001. See N_8(2n).
%H D. M. Bressoud, <a href="https://www.macalester.edu/~bressoud/books/PnC/PnCcorrect.html">Corrections: Proofs and Confirmations</a>
%H N. T. Cameron, <a href="https://www.math.hmc.edu/~cameron/dissertation.pdf">Random walks, trees and extensions of Riordan group techniques</a>, Dissertation, Howard University, 2002.
%H Naiomi Cameron and J. E. McLeod, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/McLeod/mcleod3.html">Returns and Hills on Generalized Dyck Paths</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
%H J. de Gier, <a href="https://arxiv.org/abs/math/0211285">Loops, matchings and alternating-sign matrices</a>, arXiv:math/0211285 [math.CO], 2002.
%H I. Gessel and G. Xin, <a href="https://arxiv.org/abs/math/0505217">The generating function of ternary trees and continued fractions</a>, arXiv:math/0505217 [math.CO], 2005.
%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 Yaping Liu, <a href="http://www.ripublication.com/gjpam22/gjpamv18n1_05.pdf">On the Recursiveness of Pascal Sequences</a>, Global J. of Pure and Appl. Math. (2022) Vol. 18, No. 1, 71-80.
%F 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
%F 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
%e 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.
%e 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
%p A051255 := proc(n) local i; mul((3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!),i=0..n-1); end;
%t 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 *)
%o (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
%o (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
%Y Cf. A006013, A049504.
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Michel ten Voorde_
%E Missing a(0)=1 term added by _Michael Somos_, Feb 25 2014
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