OFFSET
1,1
COMMENTS
The number of commutative discrete aggregation functions on the finite chain L_n={0,1,...,n-1,n}, i.e., the number of monotonic increasing binary functions F: L_n^2->L_n such that F(x,y)=F(y,x) for all x,y in L_n, F(0,0)=0 and F(n,n)=n.
LINKS
M. Munar, S. Massanet and D. Ruiz-Aguilera, On the cardinality of some families of discrete connectives, Information Sciences, 621 (2023), 708-728.
FORMULA
a(n) = Product_{i=1..n+1}((2i+n-1)/(2i-1) Product_{j=i+1..n+1}((i+j+n-1)/(i+j-1))) - 2*Product_{i=1..n+1}((2i+n-2)/(2i-1) Product_{j=i+1..n+1}((i+j+n-2)/(i+j-1))) + Product_{i=1..n+1}((2i+n-3)/(2i-1) Product_{j=i+1..n+1}((i+j+n-3)/(i+j-1)))
From Vaclav Kotesovec, Nov 18 2023: (Start)
a(n) = BarnesG(n+1)^(3/2) * sqrt(BarnesG(3*n)) * Gamma(3*n) * (2*(n-1) * Gamma(n/2) * Gamma(2*n)^2 - 3*2^(n+2) * Gamma(3*n/2) * Gamma(2*n) * Gamma(n+1) + 3*(3*n+1) * Gamma(n/2) * Gamma(n) * Gamma(3*n)) / (4*(2*n + 1) * BarnesG(2*n)^(3/2) * sqrt(2*Gamma(n/2) * Gamma(3*n/2)) * Gamma(2*n)^4).
a(n) ~ exp(1/24) * 3^(9*n^2/4 + 15*n/4 + 35/24) / (sqrt(A) * n^(1/24) * 2^(3*n^2 + 9*n/2 + 17/8)), where A is the Glaisher-Kinkelin constant A074962. (End)
MATHEMATICA
Table[Product[(2*i + n - 1)/(2*i - 1)*
Product[(i + j + n - 1)/(i + j - 1), {j, i + 1, n + 1}], {i, 1,
n + 1}] -
2*Product[(2*i + n - 2)/(2*i - 1)*
Product[(i + j + n - 2)/(i + j - 1), {j, i + 1, n + 1}], {i, 1,
n + 1}] +
Product[(2*i + n - 3)/(2*i - 1)*
Product[(i + j + n - 3)/(i + j - 1), {j, i + 1, n + 1}], {i, 1,
n + 1}], {n, 2, 13}]
Table[BarnesG[n+1]^(3/2) * Sqrt[BarnesG[3*n]] * Gamma[3*n] * (2*(n-1) * Gamma[n/2] * Gamma[2*n]^2 - 3*2^(n+2) * Gamma[3*n/2] * Gamma[2*n] * Gamma[n+1] + 3*(3*n+1) * Gamma[n/2] * Gamma[n] * Gamma[3*n]) / (4*(2*n + 1) * BarnesG[2*n]^(3/2) * Sqrt[2*Gamma[n/2]*Gamma[3*n/2]] * Gamma[2*n]^4), {n, 1, 14}] (* Vaclav Kotesovec, Nov 18 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Marc Munar, Oct 10 2023
STATUS
approved