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A274778
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Number of proper mergings of an n-antichain and an n-chain.
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0
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0, 3, 26, 442, 12899, 582381, 37700452, 3315996468, 380835212037, 55380159334315, 9950025870043126, 2165134468142294430, 561245519520167902471, 170913803045738754172185, 60421582956702701927410120, 24543570079301728283314502248, 11353373604627607560431407875081
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OFFSET
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0,2
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COMMENTS
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a(n) is also the number of monotone (n+1)-colorings of a complete bipartite digraph K(n,n), where a monotone (n+1)-coloring is a labeling w of the vertices of K(n,n) with integers in {1,2,...,n+1} such that for every arc (e1, e2) we have w(e1) <= w(e2).
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LINKS
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FORMULA
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a(n) = Sum_{i=1..n+1} ((n+2-i)^n - (n+1-i)^n)*i^n.
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EXAMPLE
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For n=1, the three proper mergings of a 1-chain {x} and a 1-antichain {y} are x<y, y<x, and x,y.
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MAPLE
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a := n -> add(((n-i+1)^n-(n-i)^n)*(i+1)^n, i=0..n):
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MATHEMATICA
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a[0] = 0; a[n_] := Sum[((n-i+1)^n - (n-i)^n)*(i+1)^n, {i, 0, n}];
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PROG
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(PARI) a(n) = sum(i=1, n+1, ((n+2-i)^n - (n+1-i)^n)*i^n); \\ Michel Marcus, Jul 14 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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