OFFSET
0,3
COMMENTS
The graphs of order n are properly colored from the color set {c_1, c_2,...,c_n} such that if c_i is used as a color then c_{i-1} is also used as a color.
LINKS
E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
FORMULA
Sum_{n>=0} Sum_{k>=0} T(n,k)*w^k*z^n/((1+w)^binomial(n,2)*n!) = 1/(1-(E(z,w)-1)) where E(z,w) = Sum_{n>=0} z^n/(1+w)^binomial(n,2)*n!.
EXAMPLE
Triangle begins:
1;
1;
3, 2;
13, 30, 24, 6;
75, 372, 780, 872, 546, 180, 24;
...
MATHEMATICA
nn = 8; e[z_, w_] := Sum[z^n/(n! (1 + w)^Binomial[n, 2]), {n, 0, Binomial[nn, 2]}]; Map[CoefficientList[Series[#, {w, 0, Binomial[nn, 2]}], w] &, Table[n! (1 + w)^Binomial[n, 2], {n, 0, nn}] CoefficientList[Series[1/(1 - (e[z, w] - 1)), {z, 0, nn}], z]]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Mar 12 2023
STATUS
approved