OFFSET
0,3
COMMENTS
Hankel transform of A006013 without initial term is this sequence without initial term. - 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.
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.
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
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-François 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
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Michel ten Voorde
Missing a(0)=1 term added by Michael Somos, Feb 25 2014
STATUS
approved