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A275166 Number of n-node graphs that have 2 non-isomorphic components. 10
0, 1, 1, 3, 8, 29, 140, 998, 12139, 273400, 11991356, 1018707920, 165078860603, 50500999728875, 29053989521339474, 31426435300576595334, 64000986599534312444935, 245935832697890955733422940, 1787577661113111145804012075034, 24637809007125076355873926288686728 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
"Component" means there are no edges from a node of one component to any node of the other component.
Each of the 2 components may be the empty graph with 0 nodes. That means the graph has only one "visible" component in these cases.
Each of the 2 components must be a connected graph (see A001349). (The empty graph has all properties and is a connected graph.)
The graphs of the 2 components must not be the same (not be isomorphic).
LINKS
FORMULA
G.f.: [A(x)^2 - A(x^2)]/2 where A(x) is the o.g.f. for A001349.
a(n) = A275165(n) if n odd.
EXAMPLE
a(4)=8 = 1*6 + 1*2 where 1*6=A001349(0)*A001349(4) counts graphs with an empty component and a component with 4 nodes, where 1*2 = A001349(1)*A001349(3) counts graphs with a component of 1 node and a component of 3 nodes. There is no contribution from a component of 2 nodes and another component of 2 nodes (both components were isomorphic in that case).
MATHEMATICA
terms = 20;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
A[x_] = Join[{1}, EULERi[Array[a88, terms]]].x^Range[0, terms];
(A[x]^2 - A[x^2])/2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Jan 31 2020, after Andrew Howroyd in A001349 *)
CROSSREFS
Sequence in context: A063839 A192744 A130470 * A182117 A207828 A208678
KEYWORD
nonn
AUTHOR
R. J. Mathar, Jul 18 2016
STATUS
approved

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Last modified April 24 05:49 EDT 2024. Contains 371918 sequences. (Running on oeis4.)