OFFSET
1,3
COMMENTS
a(n) is the denominator of an upper bound of the number of vertices of the polytope of the n X n X n stochastic tensors, or equivalently, of the number of Latin squares of order n, or equivalently, of the number of n X n X n line-stochastic (0,1)-tensors (see Chang et al. and Zhang et al.).
Conjecture: 1 and 3 are the only terms that appear in this sequence.
This conjecture is correct, see formula. - Kevin Ryde, Jul 01 2023
LINKS
Kevin Ryde, Table of n, a(n) for n = 1..10000
Haixia Chang, Vehbi Emrah Paksoy, and Fuzhen Zhang, Polytopes of Stochastic Tensors, Ann. Funct. Anal. 7(3): 386-393 (August 2016). arXiv:1608.03203 [math.CO], 2016. See p. 6.
Kevin Ryde, PARI/GP Code and Notes
Fuzhen Zhang and Xiao-Dong Zhang, Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors, arXiv:2110.12337 [math.CO], 2021. See p. 3.
FORMULA
A349508(n)/a(n) ~ 2^(-4 + 6*n - 6*n^2)*3^(-7/2 + 6*n - 6*n^2)*e^(-75 + 233/n + 18*n + 6*n^2)*n^(-1 - 6*n + 6*n^2)/sqrt(Pi).
a(n) = 1 if n=1 or any x[i] + y[i] >= 3 where x and y are the ternary digits of n^3 = Sum x[i]*3^i and 6*n^2 - 6*n + 3 = Sum y[i]*3^i; and a(n) = 3 otherwise. - Kevin Ryde, Jul 01 2023
MATHEMATICA
a[n_]:=Denominator[Binomial[n^3+6n^2-6n+2, n^3-1]/n^3]; Array[a, 90]
PROG
(Python)
from math import gcd, comb
def A349509(n): return n**3//gcd(comb(n*(n*(n + 6) - 6) + 2, 6*n*(n-1)+3), n**3) # Chai Wah Wu, Dec 06 2021
(PARI) \\ See links.
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Stefano Spezia, Nov 20 2021
STATUS
approved