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A381470 Number of simple connected graphs on n unlabeled nodes with exactly 2 non-overlapping cycles.

%I A381470 #6 Feb 25 2025 11:37:39
%S A381470 1,4,21,85,345,1289,4713,16622,57535,195212,653318,2158866,7063333,
%T A381470 22906699,73742762,235863378,750187968,2374249283,7481414941,
%U A381470 23482536967,73449564533,229016163367,712044375528,2208131225648,6831543467752,21089958138852,64978894444220
%N A381470 Number of simple connected graphs on n unlabeled nodes with exactly 2 non-overlapping cycles.
%C A381470 All such graphs are cactus graphs (with bridges allowed).
%H A381470 Andrew Howroyd, <a href="/A381470/b381470.txt">Table of n, a(n) for n = 6..1000</a>
%H A381470 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H A381470 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%e A381470 The a(6) = 1 graph is:
%e A381470     o       o
%e A381470    / \     / \
%e A381470   o---o---o---o
%e A381470 .
%e A381470 The a(7) = 4 graphs are:
%e A381470     o     o---o     o   o   o       o---o   o       o   o   o
%e A381470    / \    |   |    / \ / \ / \     / \     / \     / \ /   / \
%e A381470   o---o---o---o   o---o   o---o   o---o---o---o   o---o---o---o
%o A381470 (PARI) \\ TreeGf gives gf of A000081.
%o A381470 TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
%o A381470 seq(n)={my(t=TreeGf(n), t2=subst(t,x,x^2), g=t*(t^2/(1-t) + t2*(1+t)/(1-t2))/2, g2=subst(g,x,x^2)); Vec(g^2/(1-t) + g2*(1+t)/(1-t2))/2}
%Y A381470 Column k=2 of A381467.
%Y A381470 Cf. A000081, A000083, A001429.
%K A381470 nonn
%O A381470 6,2
%A A381470 _Andrew Howroyd_, Feb 25 2025