A303830 -id:A303830 - cp's OEIS frontend

A303831 Birooted graphs: number of unlabeled connected graphs with n nodes rooted at 2 indistinguishable roots.

0, 1, 3, 16, 98, 879, 11260, 230505, 7949596, 483572280, 53011686200, 10589943940654, 3880959679322754, 2623201177625659987, 3286005731275218388682, 7663042204550840483139108, 33407704152242477510352455230, 273327599183687887638526170380380

Offset: 1

Brendan McKay, May 01 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..40
Jean-François Alcover, Mathematica program

Cf. A303829 (not necessarily connected). 3rd column of A304311.

Cf. A000088 (not rooted), A126100 (connected single root), A053506 (2 roots adjacent).

Mathematica
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(* See the links section. *)
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G.f.: B(x)/G(x) - (C(x^2) + C(x)^2)/2 where B(x) is the g.f. of A303829, G(x) is the g.f. of A000088 and C(x) is the g.f. of A126100. - Andrew Howroyd, May 03 2018

a(n) = A303830(n) + A304071(n). - Brendan McKay, May 05 2018

a(12)-a(18) from Andrew Howroyd, May 03 2018

A303832 The number of edge-rooted unlabeled connected graphs with n edges.

Original entry on oeis.org

1, 1, 4, 10, 32, 101, 346, 1220, 4517, 17338, 69107, 285009, 1215015, 5344224, 24223641, 113001129, 541913075, 2668817544, 13484234188, 69831773559, 370361639587, 2009988998148, 11153858854425, 63242354288220, 366140089188603, 2163036956456422, 13031489297543608

Offset: 1

R. J. Mathar, May 04 2018

nonn

			a(1)=1: the connected graph with 1 edge (which is rooted).
a(2)=1: the connected graph with 2 edges (one rooted).
a(3)=4: the triangle graph with one choice of rooting, the linear tree with either the middle or a terminating edge rooted, the star graph with one edge rooted.

Cf. A126133 (not necessarily connected), A000664, A303830 (by number of nodes).

Cf. A002905, A339039, A339040, A339041, A339044.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1]) + G(2*n, x+A, [2]))/(2*G(2*n, x+A, [])*(1+x)))} \\ Andrew Howroyd, Nov 21 2020

G.f. A(x) satisfies: A(x)*A000664(x) = A126133(x).

A304071 Number of simple connected graphs with n nodes rooted at one non-edge.

Original entry on oeis.org

0, 0, 1, 6, 42, 402, 5381, 112776, 3935471, 240684836, 26449057257, 5289513580458, 1939502108505917, 1311274498490104492, 1642800188822966309834, 3831285832174735713684706, 16703340559932677463553709189, 136661710199022168890320488632600, 2105815888079982128884579271408161673, 61310553163194788144046000967760340771668

Offset: 1

Brendan McKay, May 05 2018

nonn

			a(3)=1: the non-edge joins the two leaves. a(4)=6: quadrangle: the non-edge is a diagonal; triangle with protruding edge: the non-edge joins the leaf with a node of degree 2; quadrangle with diagonal: the non-edge is the other diagonal; tetrahedron: no contribution; linear chain: the non-edge either joins the two leaves or a leaf with a node at distance 2; star graph: the non-edge joins two leaves.

Jean-François Alcover, Table of n, a(n) for n = 1..40

Cf. A001349 (not rooted), A126122 (not necessarily connected)

A303830 = Import["https://oeis.org/A303830/b303830.txt", "Table"][[All, 2]];
A303831 = Import["https://oeis.org/A303831/b303831.txt", "Table"][[All, 2]];
a[n_] := A303831[[n]] - A303830[[n]];
a /@ Range[1, 40] (* Jean-François Alcover, Sep 21 2019 *)

a(n) + A303830(n) = A303831(n).

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A303831 Birooted graphs: number of unlabeled connected graphs with n nodes rooted at 2 indistinguishable roots.

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A303832 The number of edge-rooted unlabeled connected graphs with n edges.

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A304071 Number of simple connected graphs with n nodes rooted at one non-edge.

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