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 A337161 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=0, k>=0. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 REFERENCES R. P. Stanley, Enumerative Combinatorics, Vol I, Second Edition, Section 3.18. LINKS FORMULA 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. EXAMPLE 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, ... MATHEMATICA 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 CROSSREFS Cf. A322280, A117402 (column k=2). Sequence in context: A210391 A071921 A003992 * A246118 A171882 A214075 Adjacent sequences:  A337158 A337159 A337160 * A337162 A337163 A337164 KEYWORD nonn,tabl AUTHOR Geoffrey Critzer, Jan 28 2021 STATUS approved

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Last modified April 15 20:33 EDT 2021. Contains 342977 sequences. (Running on oeis4.)