OFFSET
1,2
COMMENTS
Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n} satisfying the ordering principle, i.e., the number of binary functions I:L_n^2->L_n such that I is decreasing in the first argument, increasing in the second argument, I(0,0)=I(n,n)=n and I(n,0)=0 (discrete implication), and I(x,y)=n if and only if x<=y, with x,y in L_n. (ordering principle).
LINKS
Marc Munar, S. Massanet and D. Ruiz-Aguilera, On the cardinality of some families of discrete connectives, Information Sciences, Volume 621, 2023, 708-728.
FORMULA
a(n) = ((Product_{i=1..n} (2n-i)!/(n-2+i)!)*(Product_{i=2..n} Product_{j=0..i-2} (3n+1-i+2j)/2)-(Product_{i=1..n} (2n-i-1)!/(n-3+i)!)*(Product_{i=2..n} Product_{j=0..i-2} (3n-1-i+2j)/2))*(2^(n*(n-1)/2))*(Product_{i=1..n} i^(n-i)/(2n+1-2i)!).
From Vaclav Kotesovec, Nov 18 2023: (Start)
a(n) = BarnesG(n) * sqrt(BarnesG(4*n)) * Gamma(n) * (1 - Gamma(3*n-2)*Gamma(3*n-1)/(Gamma(2*n-2)*Gamma(4*n-1))) / (BarnesG(3*n) * sqrt(2*Gamma(2*n)).
a(n) ~ exp(1/24) * 2^(8*n^2 - 5*n + 1/6) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 - 3*n + 5/12)), where A is the Glaisher-Kinkelin constant A074962. (End)
MATHEMATICA
Table[(Product[Factorial[2 n - i]/Factorial[n - 2 + i], {i, 1, n}]*Product[Product[(3 n + 1 - i + 2 j)/2, {j, 0, i - 2}], {i, 2, n}] - Product[Factorial[2 n - i - 1]/Factorial[n - 3 + i], {i, 1, n}]*Product[Product[(3 n - 1 - i + 2 j)/2, {j, 0, i - 2}], {i, 2, n}])*2^((n*(n - 1))/2)*Product[i^(n - i), {i, 1, n - 1}]*Product[1/Factorial[2 n + 1 - 2 i], {i, 1, n}], {n, 1, 20}]
Table[BarnesG[n] * Sqrt[BarnesG[4*n]] * Gamma[n] * (1 - Gamma[3*n - 2]*Gamma[3*n - 1]/(Gamma[2*n - 2]*Gamma[4*n - 1])) / (BarnesG[3*n] * Sqrt[2*Gamma[2*n]]), {n, 1, 20}] (* Vaclav Kotesovec, Nov 18 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Marc Munar, Nov 18 2023
STATUS
approved