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%I #16 May 08 2023 14:29:09
%S 1,0,1,0,1,2,0,1,8,8,0,1,32,96,64,0,1,160,1152,2048,1024,0,1,1088,
%T 17920,65536,81920,32768,0,1,10368,399360,2752512,6553600,6291456,
%U 2097152,0,1,139520,13393920,168820736,692060160,1207959552,939524096,268435456
%N Triangle read by rows: T(n,k) is the number of k-colored graphs on n nodes with restricted labels, n>=0, 0<=k<=n.
%C Here, a k-colored graph on n nodes with restricted labels is a labeled k-colored graph (as in A046860) with color set {c1,c2,...,ck} such that the nodes assigned to color c1 are labeled with the integers {1,2,...,n_c1}, the nodes assigned to color c2 are labeled with the next smallest n_c2 integers {n_c1+1,n_c1+2,... n_c1+n_c2}, and generally the nodes assigned to color cj are labeled with the smallest n_cj integers not previously used to label nodes having colors c1,c2,...c(j-1) where n_cj is the number of nodes having color cj and n_c1+n_c2+...+n_ck=n and each n_cj>0.
%F Let E(x) = Sum_{n>=0} x^n/2^C(n,2). Then 1/(1-y(E(x)-1)) = Sum_{n>=0} Sum_{k=0..n} T(n,k) y^k*x^n/2^C(n,2).
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 1, 8, 8;
%e 0, 1, 32, 96, 64;
%e 0, 1, 160, 1152, 2048, 1024;
%e 0, 1, 1088, 17920, 65536, 81920, 32768;
%e ...
%t nn = 6; e[x_] := Sum[x^n/2^Binomial[n, 2],{n,0,nn}];Table[Take[(Table[2^Binomial[n, 2], {n, 0, nn}] CoefficientList[Series[1/(1 - y (e[x] - 1)), {x, 0, nn}], {x, y}])[[i]],i], {i, 1, nn + 1}] // Grid
%Y Row sums give: A335343.
%Y Cf. A046860, A006125 (main diagonal).
%K nonn,tabl
%O 0,6
%A _Geoffrey Critzer_, Jun 01 2020
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