OFFSET
3,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 3..500
Eric Weisstein's World of Mathematics, Unicyclic Graph.
Gus Wiseman, The a(6) = 25 graphs with a unique cycle.
FORMULA
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
Partial sums of A372191. - Gus Wiseman, Apr 27 2024
MATHEMATICA
Needs["Combinatorica`"]; nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
b = Drop[Flatten[
sol = SolveAlways[
0 == Series[
t[x] - x Product[1/(1 - x^i)^ a[i], {i, 1, nn}], {x, 0, nn}],
x]; Table[a[n], {n, 0, nn}] /. sol], 1];
r[x_] := Sum[b[[n]] x^n, {n, 1, nn}]; c =
Drop[Table[
CoefficientList[
Series[CycleIndex[DihedralGroup[n], s] /.
Table[s[i] -> r[x^i], {i, 1, n}], {x, 0, nn}], x], {n, 3,
nn}] // Total, 1];
d[x_] := Sum[c[[n]] x^n, {n, 1, nn}]; f =
Drop[CoefficientList[Series[r[x] - (r[x]^2 - r[x^2])/2, {x, 0, nn}],
x], 1]; Drop[CoefficientList[
Series[d[x] Product[1/(1 - x^i)^f[[i]], {i, 1, nn}], {x, 0, nn}], x], 3] (* Geoffrey Critzer, Nov 16 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jan 29 2014
EXTENSIONS
a(11)-a(30) from Geoffrey Critzer, Nov 16 2014
STATUS
approved