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A085465
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Number of monotone n-weightings of complete bipartite digraph K(3,3).
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7
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1, 15, 102, 442, 1443, 3885, 9100, 19188, 37269, 67771, 116754, 192270, 304759, 467481, 696984, 1013608, 1442025, 2011815, 2758078, 3722082, 4951947, 6503365, 8440356, 10836060, 13773565, 17346771, 21661290, 26835382, 33000927, 40304433
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OFFSET
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1,2
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COMMENTS
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A monotone n-(vertex) weighting of a digraph D=(V,E) is a function w: V -> {0,1,..,n-1} such that w(v1)<=w(v2) for every arc (v1,v2) from E.
a(n) = number of proper mergings of a 3-antichain and an (n-1)-chain. - Henri Mühle, Aug 17 2012
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LINKS
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FORMULA
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a(n) = n + 13*binomial(n, 2) + 60*binomial(n, 3) + 120*binomial(n, 4) + 108*binomial(n, 5) + 36*binomial(n, 6) = 1/20*n*(n+1)*(n^2+1)*(n^2+2*n+2) = Sum_{i=1..n} ((n+1-i)^3-(n-i)^3)*i^3. More generally, number of monotone n-weightings of complete bipartite digraph K(s, t) is Sum_{i=1..n} ((n+1-i)^s-(n-i)^s)*i^t = Sum_{i=1..n} ((n+1-i)^t-(n-i)^t)*i^s.
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MATHEMATICA
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Rest[CoefficientList[Series[x*(1 + 4*x + x^2)^2/(1 - x)^7, {x, 0, 50}], x]] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 15, 102, 442, 1443, 3885, 9100}, 40] (* Vincenzo Librandi, Oct 06 2017 *)
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PROG
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(PARI) x='x+O('x^50); Vec(x*(1+4*x+x^2)^2/(1-x)^7) \\ G. C. Greubel, Oct 06 2017
(Magma) [1/20*n*(n+1)*(n^2+1)*(n^2+2*n+2): n in [1..40]]; // Vincenzo Librandi, Oct 06 2017
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CROSSREFS
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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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