OFFSET
0,6
COMMENTS
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.
FORMULA
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).
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 1, 8, 8;
0, 1, 32, 96, 64;
0, 1, 160, 1152, 2048, 1024;
0, 1, 1088, 17920, 65536, 81920, 32768;
...
MATHEMATICA
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Jun 01 2020
STATUS
approved