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A236570 Number of n-node simple unicyclic graphs. 3
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 (list; graph; refs; listen; history; text; internal format)
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
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
Cf. A001429 (number of connected n-node unicyclic graphs).
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

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Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)