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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 implies C(v_i) <= C(v_j); n>=0, k>=0.

%I A337161 #19 Jan 28 2021 21:42:03
%S A337161 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,
%T A337161 84,195,162,1,0,1,7,36,165,644,1635,1090,1,0,1,8,49,286,1605,7620,
%U A337161 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
%N 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<j implies C(v_i) <= C(v_j); n>=0, k>=0.
%D A337161 R. P. Stanley, Enumerative Combinatorics, Vol I, Second Edition, Section 3.18.
%F A337161 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.
%e A337161   1, 1,    1,     1,      1,      1,       1, ...
%e A337161   0, 1,    2,     3,      4,      5,       6, ...
%e A337161   0, 1,    4,     9,     16,     25,      36, ...
%e A337161   0, 1,   10,    35,     84,    165,     286, ...
%e A337161   0, 1,   34,   195,    644,   1605,    3366, ...
%e A337161   0, 1,  162,  1635,   7620,  24389,   62310, ...
%e A337161   0, 1, 1090, 21187, 143748, 599685, 1882054, ...
%t A337161 nn = 6; e[x_] := Sum[x^n/(2^Binomial[n, 2]), {n, 0, nn}];
%t A337161 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
%Y A337161 Cf. A322280, A117402 (column k=2).
%K A337161 nonn,tabl
%O A337161 0,8
%A A337161 _Geoffrey Critzer_, Jan 28 2021