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A214650 Number of distinct connected unicyclic bipartite graphs with n vertices.

%I A214650 #12 Feb 16 2025 08:33:18
%S A214650 0,0,0,1,1,5,10,34,85,254,690,1997,5582,15975,45244,129254,368215,
%T A214650 1052961,3010169,8622273,24709964,70902886,203594559,585163116,
%U A214650 1683079071,4844758076,13955265122,40225474849,116021495035,334843170810,966929417619,2793756318793
%N A214650 Number of distinct connected unicyclic bipartite graphs with n vertices.
%C A214650 The graphs also have n edges.
%H A214650 Andrew Howroyd, <a href="/A214650/b214650.txt">Table of n, a(n) for n = 1..200</a>
%H A214650 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnicyclicGraph.html">Unicyclic Graph</a>
%e A214650 a(5)=1, a 4-cycle plus a pendant edge.
%o A214650 (PARI) \\ TreeGf gives gf of A000081
%o A214650 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 A214650 seq(n)={concat([0,0,0], if(n<4, [], my(t=TreeGf(n-2)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(sum(k=2, n\2, sumdiv(2*k, d, eulerphi(d)*g(d)^(2*k/d))/k + (g(1)^2+g(2))*g(2)^(k-1))/4)))} \\ _Andrew Howroyd_, May 22 2018
%Y A214650 Cf. A001429.
%K A214650 nonn
%O A214650 1,6
%A A214650 _David Bevan_, Jul 24 2012
%E A214650 Terms a(16) and beyond from _Andrew Howroyd_, May 22 2018