A335343 Number of k-colored graphs on n nodes with restricted labels.

1, 1, 3, 17, 193, 4385, 199233, 18104449, 3290333441, 1195981275649, 869438472061953, 1264105507046557697, 3675850064599476867073, 21377762572680129683660801, 248654719090254548473238011905, 5784437834927690918603693712506881

Offset: 0

Author

Geoffrey Critzer, Jun 02 2020

Keywords

• nonn

Comments

A k-colored graph on n nodes with restricted labels is a labeled k-colored graph (as in A334282) 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 ncj is the number of nodes having color j and nc1+nc2+...+nck=n and each ncj>0.

Crossrefs

Row sums of A335330.

Cf. A334282.

Programs

Mathematica

nn = 15; e[x_] := Sum[x^n/2^Binomial[n, 2], {n, 0, nn}]; Table[2^Binomial[n, 2], {n, 0, nn}] CoefficientList[Series[1/(1 - (e[x] - 1)), {x, 0, nn}], x]

Formula

Let E(x)=Sum_n>=0 x^n/2^C(n,2). Then 1/(1-(E(x)-1)) = Sum_n>=0 a(n)*x^n/2^C(n,2).