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A367541
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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.
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0
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1, 9, 246, 21307, 5967884, 5464753020, 16464650143150, 163867734760669875, 5401439489386802569500, 590665306641885854720733600, 214530897918187139967720562273920, 258998339526821950480574606267461843536, 1039917052871541256867935621512668719049634384, 13891789744852831118958512413787919060197070057215380
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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)!).
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
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MATHEMATICA
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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}]
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CROSSREFS
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Particular case of the enumeration of discrete implications in general, enumerated in A360612.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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