A340811 -id:A340811 - cp's OEIS frontend

A054581 Number of unlabeled 2-trees with n nodes.

0, 1, 1, 1, 2, 5, 12, 39, 136, 529, 2171, 9368, 41534, 188942, 874906, 4115060, 19602156, 94419351, 459183768, 2252217207, 11130545494, 55382155396, 277255622646, 1395731021610, 7061871805974, 35896206800034, 183241761631584

Offset: 1

Vladeta Jovovic, Apr 11 2000

A 2-tree is recursively defined as follows: K_2 is a 2-tree and any 2-tree on n+1 vertices is obtained by joining a vertex to a 2-clique in a 2-tree on n vertices. Care is needed with the term 2-tree (and k-tree in general) because it has at least two commonly used definitions.
A036361 gives the labeled version of this sequence, which has an easy formula analogous to Cayley's formula for the number of trees.
Also, number of unlabeled 3-gonal 2-trees with n 3-gons.

			a(1)=0 because K_1 is not a 2-tree;
a(2)=a(3)=1 because K_2 and K_3 are the only 2-trees on those sizes.
a(4)=1 because there is a unique example obtained by joining a triangle to K_3 along an edge (thus forming K_4\e). The two graphs on 5 nodes are obtained by joining a triangle to K_4\e, either along the shared edge or along one of the non-shared edges.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 327-328.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, t(x), (3.5.19).

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
T. Fowler, I. Gessel, G. Labelle, and P. Leroux, The specification of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, Table 1.
Nick Early, Anaëlle Pfister, and Bernd Sturmfels, Minimal Kinematics on M_{0,n}, arXiv:2402.03065 [math.AG], 2024.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45. - From _N. J. A. Sloane_, Dec 15 2012
Gilbert Labelle, Cédric Lamathe, and Pierre Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], 2003.
Eric Weisstein's World of Mathematics, k-Tree.
Index entries for sequences related to trees

Column k=3 of A340811, column k=2 of A370770.

Cf. A000272 (labeled trees), A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees).

Additional comments from Gordon F. Royle, Dec 02 2002

Missing initial term 0 inserted by Brendan McKay, Aug 07 2023

A340812 Array read by antidiagonals: T(n,k) is the number of unlabeled oriented k-gonal 2-trees with n oriented polygons, n >= 0, k >= 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 3, 7, 6, 1, 1, 1, 3, 11, 18, 11, 1, 1, 1, 4, 17, 49, 68, 23, 1, 1, 1, 4, 25, 96, 252, 251, 47, 1, 1, 1, 5, 33, 177, 687, 1406, 1020, 106, 1, 1, 1, 5, 43, 285, 1537, 5087, 8405, 4258, 235, 1, 1, 1, 6, 55, 442, 3014, 14310, 40546, 52348, 18580, 551

Offset: 0

Andrew Howroyd, Feb 02 2021

See section 3 of the Labelle reference.

			Array begins:
=========================================================
n\k |  2    3    4     5      6      7      8       9
----+----------------------------------------------------
  0 |  1    1    1     1      1      1      1       1 ...
  1 |  1    1    1     1      1      1      1       1 ...
  2 |  1    1    1     1      1      1      1       1 ...
  3 |  2    2    3     3      4      4      5       5 ...
  4 |  3    7   11    17     25     33     43      55 ...
  5 |  6   18   49    96    177    285    442     635 ...
  6 | 11   68  252   687   1537   3014   5370    8901 ...
  7 | 23  251 1406  5087  14310  33632  70000  132533 ...
  8 | 47 1020 8405 40546 141582 399065 966254 2089103 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.

Columns 2..4 are A000055, A303742, A340813.

Cf. A340811 (unoriented case), A340814 (edge-rooted case).

\\ here B(n,k) gives column k of A340814.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(n, k)={my(p=1+O(x)); for(n=1, n, p=1+x*Ser(EulerT(Vec(p^(k-1))))); p}
C(n, k)={my(p=B(n,k)); Vec(p - x*p^k + x*sumdiv(k, d, eulerphi(d)*subst(p + O(x*x^(n\d)), x, x^d)^(k/d))/k)}
{ Mat(vector(7, k, C(7, k+1)~)) }

G.f. of column k: B(x) - x*B(x)^k + x*(Sum_{d|k} phi(d)*B(x^d)^(k/d))/k, where B(x) if the g.f. of column k of A340814.

A094610 Number of unlabeled 4-gonal 2-trees with n 4-gons.

Original entry on oeis.org

1, 1, 1, 3, 8, 32, 141, 749, 4304, 26492, 169263, 1115015, 7507211, 51466500, 358100288, 2523472751, 17978488711, 129325796854, 938234533024, 6858551493579, 50478955083341, 373815860588360, 2783730088637429, 20835514668765200, 156675846789962554, 1183195305751233833

Offset: 0

Vladeta Jovovic, Jun 06 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees

Column k=4 of A340811.

Cf. A054581.

Terms a(21) and beyond from Andrew Howroyd, May 09 2021

A094611 Number of unlabeled pentagonal 2-trees with n pentagons.

Original entry on oeis.org

1, 1, 1, 3, 11, 56, 359, 2597, 20386, 167819, 1429815, 12500748, 111595289, 1013544057, 9340950309, 87176935700, 822559721606, 7836316493485, 75293711520236, 728968295958626, 7105984356424859, 69697524828651997, 687448151462822966, 6815174949649428254

Offset: 0

Vladeta Jovovic, Jun 06 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees

Column k=5 of A340811.

Cf. A054581.

Terms a(21) and beyond from Andrew Howroyd, May 09 2021

A094637 Number of unlabeled hexagonal 2-trees with n hexagons.

Original entry on oeis.org

1, 1, 1, 4, 16, 103, 799, 7286, 71094, 729974, 7743818, 84307887, 937002302, 10595117272, 121568251909, 1412555701804, 16594126114458, 196829590326284, 2354703777373055, 28385225424840078, 344524656398655124, 4207569734885041198, 51674558137433627724

Offset: 0

Vladeta Jovovic, Jun 06 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.

Column k=6 of A340811.

Cf. A054581.

Terms a(21) and beyond from Andrew Howroyd, Feb 02 2021

A094651 Number of unlabeled heptagonal 2-trees with n heptagons.

Original entry on oeis.org

1, 1, 1, 4, 20, 158, 1539, 16970, 199879, 2460350, 31266165, 407461893, 5420228329, 73352481577, 1007312969202, 14008437540003, 196963172193733, 2796235114720116, 40038505601111596, 577693117173844307, 8392528734991449808, 122680472402091278920, 1803416089912665770579

Offset: 0

Vladeta Jovovic, Jun 06 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.

Column k=7 of A340811.

Cf. A054581.

Terms a(20) and beyond from Andrew Howroyd, Feb 02 2021

A094652 Number of unlabeled octagonal 2-trees with n octagons.

Original entry on oeis.org

1, 1, 1, 5, 26, 245, 2737, 35291, 483819, 6937913, 102666626, 1558022255, 24133790815, 380320794122, 6081804068869, 98490990290897, 1612634990857755, 26660840123167203, 444560998431678554, 7469779489114328514, 126375763235359105446, 2151342943131421629636

Offset: 0

Vladeta Jovovic, Jun 06 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.

Column k=8 of A340811.

Cf. A054581.

Terms a(20) and beyond from Andrew Howroyd, Feb 02 2021

A094653 Number of unlabeled 9-gonal 2-trees with n 9-gons.

Original entry on oeis.org

1, 1, 1, 5, 32, 343, 4505, 66603, 1045335, 17115162, 289107854, 5007144433, 88516438360, 1591949961503, 29053438148676, 536972307386326, 10034276171127780, 189331187319203010, 3603141751525175854, 69097496637591215442, 1334213677527481808220, 25922717218623692348237

Offset: 0

Vladeta Jovovic, Jun 06 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.

Column k=9 of A340811.

Cf. A054581.

Terms a(19) and beyond from Andrew Howroyd, Feb 02 2021

A094654 Number of unlabeled 10-gonal 2-trees with n 10-gons.

Original entry on oeis.org

1, 1, 1, 6, 39, 482, 7053, 117399, 2070289, 38097139, 723169329, 14074851642, 279609377638, 5651139037570, 115901006038377, 2407291353219949, 50553753543016719, 1071971262516091572, 22926544048209731554, 494103705426160765546, 10722146465907412669810

Offset: 0

Vladeta Jovovic, Jun 06 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.

Column k=10 of A340811.

Cf. A054581.

Terms a(19) and beyond from Andrew Howroyd, Feb 02 2021

A094655 Number of unlabeled 11-gonal 2-trees with n 11-gons.

Original entry on oeis.org

1, 1, 1, 6, 46, 636, 10527, 194997, 3823327, 78118107, 1646300388, 35570427615, 784467060622, 17601062294302, 400750115756742, 9240636709048733, 215435023547580882, 5071520482516388865, 120417032326341878672, 2881134828445365441407, 69410468220307148620226

Offset: 0

Vladeta Jovovic, Jun 06 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.

Column k=11 of A340811.

Cf. A054581.

Terms a(19) and beyond from Andrew Howroyd, Feb 02 2021

cp's OEIS Frontend

A054581 Number of unlabeled 2-trees with n nodes.

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A340812 Array read by antidiagonals: T(n,k) is the number of unlabeled oriented k-gonal 2-trees with n oriented polygons, n >= 0, k >= 2.

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A094610 Number of unlabeled 4-gonal 2-trees with n 4-gons.

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A094611 Number of unlabeled pentagonal 2-trees with n pentagons.

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A094637 Number of unlabeled hexagonal 2-trees with n hexagons.

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A094651 Number of unlabeled heptagonal 2-trees with n heptagons.

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A094652 Number of unlabeled octagonal 2-trees with n octagons.

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A094653 Number of unlabeled 9-gonal 2-trees with n 9-gons.

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A094654 Number of unlabeled 10-gonal 2-trees with n 10-gons.

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A094655 Number of unlabeled 11-gonal 2-trees with n 11-gons.

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