A005470 -id:A005470 - cp's OEIS frontend

A038035 Number of labeled dyslexic planted planar trees with n+1 nodes.

1, 2, 9, 72, 840, 12780, 238770, 5281920, 134946000, 3909578400, 126638542800, 4535037460800, 177904622095200, 7586967310322400, 349479111223242000, 17292052928037888000, 914673660594613920000, 51506610632458293312000, 3076341001739003430432000

Offset: 1

Christian G. Bower, Sep 15 1998

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to rooted trees

Cf. A005470, A032128.

m = 20;
CoefficientList[InverseSeries[2*x*(1 - x)/(2 - x^2) + O[x]^m], x]*Range[0, m - 1]! // Rest (* Jean-François Alcover, Sep 08 2019 *)

Vec(serlaplace(serreverse(2*x*(1 - x)/(2 - x^2) + O(x^20)))) \\ Andrew Howroyd, Sep 19 2018

Divides by n and shifts left under "BIJ" (reversible, indistinct, labeled) transform.

E.g.f.: series reversion of 2*x*(1 - x)/(2 - x^2). - Andrew Howroyd, Sep 19 2018

Terms a(16) and beyond from Andrew Howroyd, Sep 19 2018

A112948 Number of unrooted 3-regular planar maps with 2n vertices, up to orientation-preserving isomorphisms.

Original entry on oeis.org

2, 6, 26, 191, 1904, 22078, 282388, 3848001, 54953996, 814302292

Offset: 1

Valery A. Liskovets, Oct 10 2005

nonn

A 3-regular map is a regular map with valency 3.

			There exist 2 planar maps with two 3-valent vertices: a map with three parallel edges and a map with one loop in each vertex and a link connecting the vertices. Therefore a(1)=2.

Z. C. Gao, V. A. Liskovets and N. C. Wormald, Enumeration of unrooted odd-valent regular planar maps, Preprint, 2005.
Mark van Hoeij, Vijay Jung Kunwar, Classifying (near)-Belyi maps with Five Exceptional Points, arXiv preprint arXiv:1604.08158, 2016. Also in Indagationes Mathematicae (2019) Vol. 30, No. 1, 136-156.
Riccardo Murri, Fatgraph algorithms and the homology of the Kontsevich complex, arXiv preprint arXiv:1202.1820, 2012.

Cf. A112944, A112945, A112949 (5-regular), A005470.

3-regular maps on the torus: A292408.

A145269 Number of simple nonplanar graphs on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 14, 222, 5380, 194815, 10864252, 1000316856, 164757860141

Offset: 0

Eric W. Weisstein, Oct 05 2008

Slavik Jablan, Ljiljana Radovic, Radmila Sazdanovic (2011). Nonplanar graphs derived from Gauss codes of virtual knots and links, J. Math. Chem. 49(10), 2250-2267, DOI:10.1007/s10910-011-9884-6
Eric Weisstein's World of Mathematics, Nonplanar Graph

Cf. A145270.

a(n) = A000088(n) - A005470(n).

Edited by Max Alekseyev, Sep 18 2009

a(12) added by Max Alekseyev, Aug 17 2015

A307071 Number of simple graphs on n nodes with crossing number 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 12, 162, 3183, 74696, 1892122

Offset: 1

Eric W. Weisstein, Mar 22 2019

			5-node: K_5.
6-node: includes K_{1,2,3}, K_5+K_1, (5-1)-lollipop, 2 X 3 queen, utility graph K_{3,3}.

Cf. A005470 (graphs with crossing number 0).

Cf. A307072 (connected graphs with crossing number 1).

a(9) from Eric W. Weisstein, Apr 17 2019

a(10) from Eric W. Weisstein, Apr 28 2019

A003584 Unicursal (i.e., possessing an Eulerian path) planar rooted maps with n edges.

Original entry on oeis.org

1, 2, 9, 52, 336, 2304, 16368, 118976, 878592, 6562816, 49447424, 375072768, 2860343296, 21909012480, 168425533440, 1298753372160, 10041201131520, 77809145610240, 604138825973760, 4698956908462080, 36604934482821120

Offset: 0

Valery A. Liskovets

Unicursal (in a broad sense) means that no more than two vertices are of odd valency (that is maps possessing an Eulerian path or cycle).

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A000257, A005470, A069720.

a[n_] := 2^(n-1)*(3*Binomial[2*n, n]/((n+1)*(n+2))+Binomial[2*n-1, n]); a[0]=1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 11 2014 *)

a(n) = A000257(n) + A069720(n).

More terms from Valery A. Liskovets, Apr 07 2002

A049369 Number of n-node planar graphs with minimum degree at least 1.

Original entry on oeis.org

0, 1, 2, 7, 22, 109, 680, 6144, 72887, 1061063, 17540092, 314631443

Offset: 1

Brendan McKay

Cf. A005470.

a(12) from Sean A. Irvine, Aug 13 2021

A069732 Number of nonisomorphic unrooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

0, 0, 2, 7, 40, 239, 1549, 10396, 71467, 498598, 3520015, 25087426, 180249182, 1304148015, 9494015372, 69492950976, 511147940104, 3776180492129, 28007532925171, 208474866181148

Offset: 1

Valery A. Liskovets, Apr 07 2002

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069724, A003645, A069725, A005470.

A003645[n_] := 2^n CatalanNumber[n + 1];
A069724[n_] := 1/(2 n) DivisorSum[n, If[OddQ[n/#], EulerPhi[n/#] 2^(# - 2) Binomial[2 #, #], 0] &] + If[OddQ[n], 2^((n - 3)/2) Binomial[n - 1, (n - 1)/2], 2^((n - 6)/2) Binomial[n, n/2]];
A069725[n_] := If[n <= 2, 1, With[{m = Floor[(n + 1)/2]}, 1/n 2^(n - 3) Binomial[2 n - 2, n - 1] + 2^(m - 3) Binomial[2 m - 2, m - 1]]];
a[n_] := If[n == 1, 0, A069724[n] - A003645[n - 2] - A069725[n]];
Array[a, 20] (* Jean-François Alcover, Aug 28 2019 *)

a(n) = A069724(n) - A003645(n) - A069725(n).

A119501 Number of isomorphism classes of 3-connected simple planar graphs (convex polytopes) where isomorphism does not allow reflection.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 45, 419, 4798, 62754, 872411, 12728018, 192324654, 2991463239, 47663036427, 775158142233, 12831576165782

Offset: 1

Brendan McKay, Jun 02 2006

If reflection is allowed, the counts are A000944.

Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs

Cf. A005470.

A173422 Partials sums of A003094 (unlabeled connected planar simple graphs with n nodes).

Original entry on oeis.org

1, 2, 3, 5, 11, 31, 130, 776, 6750, 78635, 1131440, 18580739, 331953037

Offset: 0

Jonathan Vos Post, Feb 18 2010

Partials sums of number of unlabeled connected planar simple graphs with n nodes. The subsequence of primes in these partial sums begins: 2, 3, 5, 11, 31.

			a(11) = 1 + 1 + 1 + 2 + 6 + 20 + 99 + 646 + 5974 + 71885 + 1052805 + 17449299.

Cf. A003094, inverse Euler transform of A005470, A126201.

S=0; A173422=apply(t->S+=t,A003094) \\ with A003094 defined as vector of the initial terms of that sequence. - M. F. Hasler, Mar 20 2018

a(n) = Sum_{i=0..n} A003094(i).

Edited and a(12) added by M. F. Hasler, Mar 20 2018

A298445 Triangle T(n,k) read by rows: number of n-node simple graphs with rectilinear crossing number k (k=0..A014540(n)).

Original entry on oeis.org

1, 2, 4, 11, 33, 1, 142, 12, 1, 1, 822, 162, 39, 16, 1, 2, 1, 0, 0, 1, 6966, 3183, 1291, 559, 172, 82, 48, 12, 15, 8, 4, 1, 3, 0, 0, 1, 0, 0, 0, 1, 79853

Offset: 1

Eric W. Weisstein, Jan 19 2018

Computed up to n=8 using data provided by Geoffrey Exoo. (There appear to be some problems with n=9 data.)

			Triangle begins:
1
2
4
11
33, 1
142, 12, 1, 1
822, 162, 39, 16, 1, 2, 1, 0, 0, 1
6966, 3183, 1291, 559, 172, 82, 48, 12, 15, 8, 4, 1, 3, 0, 0, 1, 0, 0, 0, 1

Eric Weisstein's World of Mathematics, Rectilinear Crossing Number
Eric Weisstein's World of Mathematics, Simple Graph

Cf. A014540 (rectilinear crossing number for K_n).
Cf. A298446 (counts for simple connected graphs).
Cf. A307071 (number of simple graphs with crossing number 1).

T(n,0) = A005470(n).
T(n,1) = A307071(n).
kmax(n) = A014540(n).
T(n,kmax(n)) = 1 for n > 4.
Sum_{k=0..kmax(n)} T(n,k) = A000088(n).

Corrected by Eric W. Weisstein, Mar 28 2019

cp's OEIS Frontend

A038035 Number of labeled dyslexic planted planar trees with n+1 nodes.

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A112948 Number of unrooted 3-regular planar maps with 2n vertices, up to orientation-preserving isomorphisms.

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A145269 Number of simple nonplanar graphs on n nodes.

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A307071 Number of simple graphs on n nodes with crossing number 1.

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A003584 Unicursal (i.e., possessing an Eulerian path) planar rooted maps with n edges.

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A049369 Number of n-node planar graphs with minimum degree at least 1.

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A069732 Number of nonisomorphic unrooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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A119501 Number of isomorphism classes of 3-connected simple planar graphs (convex polytopes) where isomorphism does not allow reflection.

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A173422 Partials sums of A003094 (unlabeled connected planar simple graphs with n nodes).

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A298445 Triangle T(n,k) read by rows: number of n-node simple graphs with rectilinear crossing number k (k=0..A014540(n)).