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%I #5 Jan 11 2021 23:08:42
%S 1,1,2,7,50,520,10665,400220,29204589,4143245857,1146827743079,
%T 619412332805088,653237982066620540,1346571060160843394520,
%U 5432476352054378478159877,42947950068987980977264834190,666212968663987333085874313873428,20301440661023158546856805172595805762
%N Number of inequivalent vertex colorings of connected graphs on n unlabeled vertices.
%C Equivalence is up to permutation of the colors. Adjacent vertices may have the same color.
%e a(3) = 7 because there are 2 connected graphs on 3 vertices. The complete graph K_3 can be coloring in 3 ways (111, 112, 123) and the path graph P_3 can be colored in 4 ways (111, 112, 121, 123).
%o (PARI) \\ See links in A339645 for combinatorial species functions.
%o edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
%o graphsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 2^edges(p) * sMonomial(p)); s/n!}
%o graphsSeries(n)={sum(k=0, n, graphsCycleIndex(k)*x^k) + O(x*x^n)}
%o InequivalentColoringsSeq(1+sLog(graphsSeries(15)))
%Y Cf. A340024, A340025, A340026.
%K nonn
%O 0,3
%A _Andrew Howroyd_, Jan 02 2021
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