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%I #26 Aug 01 2024 12:05:21
%S 1,3,9,25,68,185,504,1379,3788,10480,29094,81193,227379,639099,
%T 1801394,5091388,14422301,40939337,116420959,331622137,946020596,
%U 2702412657,7729367873,22132856218,63444473053,182046034559,522841943138,1502920139133
%N Number of n-node simple unicyclic graphs.
%H Andrew Howroyd, <a href="/A236570/b236570.txt">Table of n, a(n) for n = 3..500</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnicyclicGraph.html">Unicyclic Graph</a>.
%H Gus Wiseman, <a href="/A236570/a236570.png">The a(6) = 25 graphs with a unique cycle</a>.
%F G.f.: A(x)*B(x) where A(x) is the o.g.f. for A001429 and B(x) is the o.g.f. for A005195. - _Geoffrey Critzer_, Nov 16 2014
%F Partial sums of A372191. - _Gus Wiseman_, Apr 27 2024
%t Needs["Combinatorica`"];nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
%t b = Drop[Flatten[
%t sol = SolveAlways[
%t 0 == Series[
%t t[x] - x Product[1/(1 - x^i)^ a[i], {i, 1, nn}], {x, 0, nn}],
%t x]; Table[a[n], {n, 0, nn}] /. sol], 1];
%t r[x_] := Sum[b[[n]] x^n, {n, 1, nn}]; c =
%t Drop[Table[
%t CoefficientList[
%t Series[CycleIndex[DihedralGroup[n], s] /.
%t Table[s[i] -> r[x^i], {i, 1, n}], {x, 0, nn}], x], {n, 3,
%t nn}] // Total, 1];
%t d[x_] := Sum[c[[n]] x^n, {n, 1, nn}]; f =
%t Drop[CoefficientList[Series[r[x] - (r[x]^2 - r[x^2])/2, {x, 0, nn}],
%t x], 1]; Drop[CoefficientList[
%t Series[d[x] Product[1/(1 - x^i)^f[[i]], {i, 1, nn}], {x, 0, nn}], x],3] (* _Geoffrey Critzer_, Nov 16 2014 *)
%Y The covering version is A372191, labeled A372195.
%Y The labeled version is A372193.
%Y Cf. A001429 (number of connected n-node unicyclic graphs), A005195.
%K nonn
%O 3,2
%A _Eric W. Weisstein_, Jan 29 2014
%E a(11)-a(30) from _Geoffrey Critzer_, Nov 16 2014
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