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%I #13 Dec 17 2023 11:21:37
%S 1,9,246,21307,5967884,5464753020,16464650143150,163867734760669875,
%T 5401439489386802569500,590665306641885854720733600,
%U 214530897918187139967720562273920,258998339526821950480574606267461843536,1039917052871541256867935621512668719049634384,13891789744852831118958512413787919060197070057215380
%N Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the identity principle, i.e., I(x,x)=n for all x in L_n.
%C Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n} satisfying the identity 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,x)=n for all x in L_n (identity principle).
%H Marc Munar, S. Massanet, and D. Ruiz-Aguilera, <a href="https://doi.org/10.1016/j.ins.2022.10.121">On the cardinality of some families of discrete connectives</a>, Information Sciences, Volume 621, 2023, 708-728.
%F a(n) = ((Product_{i=1..n} (2n-i+1)!/(n-1+i)!)*(Product_{i=2..n} Product_{j=0..i-2} (3n+3-i+2j)/2)-(Product_{i=1..n} (2n-i)!/(n-2+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)!).
%F a(n) ~ exp(1/24) * 2^(2/3 + 5*n + 8*n^2) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 + 3*n + 5/12)), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Nov 29 2023
%t Table[(Product[Factorial[2 n - i + 1]/Factorial[n - 1 + i], {i, 1, n}]*Product[Product[(3 n + 3 - i + 2 j)/2, {j, 0, i - 2}], {i, 2, n}] - 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}])*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, 15}]
%Y Particular case of the enumeration of discrete implications in general, enumerated in A360612.
%Y Cf. A074962.
%K nonn,easy
%O 1,2
%A _Marc Munar_, Nov 22 2023
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