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A236570
Number of n-node simple unicyclic graphs.
14
1, 3, 9, 25, 68, 185, 504, 1379, 3788, 10480, 29094, 81193, 227379, 639099, 1801394, 5091388, 14422301, 40939337, 116420959, 331622137, 946020596, 2702412657, 7729367873, 22132856218, 63444473053, 182046034559, 522841943138, 1502920139133
OFFSET
3,2
LINKS
Eric Weisstein's World of Mathematics, Unicyclic Graph.
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
The covering version is A372191, labeled A372195.
The labeled version is A372193.
Cf. A001429 (number of connected n-node unicyclic graphs), A005195.
Sequence in context: A094292 A295571 A291019 * A338726 A323362 A201533
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jan 29 2014
EXTENSIONS
a(11)-a(30) from Geoffrey Critzer, Nov 16 2014
STATUS
approved