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A334282 Number of properly colored labeled graphs on n nodes so that the color function is surjective onto {c_1,c_2,...,c_k} for some k, 1<=k<=n.

%I A334282 #14 Dec 12 2023 17:55:08
%S A334282 1,1,5,73,2849,277921,65067905,35545840513,44384640206849,
%T A334282 124697899490480641,778525887500557625345,10693248499002776513697793,
%U A334282 320453350845793018626300755969,20807125028666778079876193487790081,2909872870574162514727072641529432735745
%N A334282 Number of properly colored labeled graphs on n nodes so that the color function is surjective onto {c_1,c_2,...,c_k} for some k, 1<=k<=n.
%C A334282 Also 1 together with the row sums of A046860.
%C A334282 A binary relation R on [n] is periodic iff there is a d>=2 such that R^d = R.  Let A be the class of non-arcless strongly connected periodic relations (A000629).  Then a(n) is the number of binary relations on [n] whose strongly connected components are in A. - _Geoffrey Critzer_, Dec 12 2023
%H A334282 Alois P. Heinz, <a href="/A334282/b334282.txt">Table of n, a(n) for n = 0..77</a>
%F A334282 Sum_{n>=0} a_n*x^n/(n!*2^C(n,2)) = 1/(2-Sum_{n>=0} x^n/(n!*2^C(n,2))).
%p A334282 b:= proc(n, k) option remember; `if`([n, k]=[0$2], 1,
%p A334282       add(binomial(n, r)*2^(r*(n-r))*b(r, k-1), r=0..n-1))
%p A334282     end:
%p A334282 a:= n-> add(b(n,k), k=0..n):
%p A334282 seq(a(n), n=0..15);  # _Alois P. Heinz_, Apr 21 2020
%t A334282 nn = 15; e2[x_] := Sum[x^n/(n! 2^Binomial[n, 2]), {n, 0, nn}];
%t A334282 Table[n! 2^Binomial[n, 2], {n, 0, nn}] CoefficientList[Series[1/(1 - (e2[x] - 1)), {x, 0, nn}], x]
%Y A334282 Cf. A046860, A322280, A011266, A361560, A003024, A365534, A365590.
%K A334282 nonn
%O A334282 0,3
%A A334282 _Geoffrey Critzer_, Apr 21 2020