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A337161
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Square array read by antidiagonals: T(n,k) is the number of simple labeled graphs G with vertex set V(G) = {v_1,...,v_n} along with a (coloring) function C:V(G) ->[k] such that v_i adjacent to v_j implies C(v_i) != C(v_j) and i<j implies C(v_i) <= C(v_j); n>=0, k>=0.
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0
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 10, 1, 0, 1, 5, 16, 35, 34, 1, 0, 1, 6, 25, 84, 195, 162, 1, 0, 1, 7, 36, 165, 644, 1635, 1090, 1, 0, 1, 8, 49, 286, 1605, 7620, 21187, 10370, 1, 0, 1, 9, 64, 455, 3366, 24389, 143748, 430467, 139522, 1, 0, 1, 10, 81, 680, 6279, 62310, 599685, 4412164, 13812483, 2654722, 1, 0, 1, 11, 100, 969, 10760, 136871, 1882054, 24413445, 223233540, 702219779, 71435266, 1, 0
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OFFSET
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0,8
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Vol I, Second Edition, Section 3.18.
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LINKS
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FORMULA
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Let e(x) = Sum_{n>=0} x^n/2^binomial(n,2). Then e(x)^k = Sum_{n>=0} Z_n(k)*x^n/2^biomial(n,2) and T(n,k) = Z_n(k). Z_n(k) is the zeta polynomial of the class of posets described in A117402.
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EXAMPLE
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1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 4, 9, 16, 25, 36, ...
0, 1, 10, 35, 84, 165, 286, ...
0, 1, 34, 195, 644, 1605, 3366, ...
0, 1, 162, 1635, 7620, 24389, 62310, ...
0, 1, 1090, 21187, 143748, 599685, 1882054, ...
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MATHEMATICA
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nn = 6; e[x_] := Sum[x^n/(2^Binomial[n, 2]), {n, 0, nn}];
Table[Table[2^Binomial[n, 2], {n, 0, nn}] PadRight[CoefficientList[Series[e[x]^k, {x, 0, nn}], x], nn + 1], {k, 0, nn}] // Transpose // Grid
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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