
COMMENTS

Partial sums of A245883. This may be proved using two facts: (i) the number of connected components of a graph is the multiplicity of the root 0 of the chromatic polynomial (thus the chromatic polynomial determines whether a graph is connected) and (ii) a disconnected graph is chromatically equivalent to some graph with an isolated vertex. The first statement is well known. For the latter statement, see p. 65 of [Dong].  Eric M. Schmidt, Mar 20 2015
A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic polynomial is given by chi_G(x) = Sum_p (x)_k, where the sum is over all stable partitions of G, k is the length (number of blocks) of p, and (x)_k is the falling factorial x(x1)(x2)...(xk+1).  Gus Wiseman, Nov 24 2018


MATHEMATICA

spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}];
falling[x_, k_]:=Product[(xi), {i, 0, k1}];
chromPoly[g_]:=Expand[Sum[falling[x, Length[stn]], {stn, spsu[Select[Subsets[Union@@g], Select[DeleteCases[g, {_}], Function[ed, Complement[ed, #]=={}]]=={}&], Union@@g]}]];
simpleSpans[n_]:=simpleSpans[n]=If[n==0, {{}}, Union@@Table[If[#=={}, Union[ine, {{n}}], Union[Complement[ine, List/@#], {#, n}&/@#]]&/@Subsets[Range[n1]], {ine, simpleSpans[n1]}]];
Table[Length[Union[chromPoly/@simpleSpans[n]]], {n, 5}] (* Gus Wiseman, Nov 24 2018 *)
