A054581 -id:A054581 - cp's OEIS frontend

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

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

A094656 Number of unlabeled 12-gonal 2-trees with n 12-gons.

Original entry on oeis.org

1, 1, 1, 7, 55, 840, 15189, 309607, 6671842, 149850849, 3471296793, 82442359291, 1998559329142, 49290785442796, 1233639304644946, 31268489727956101, 801335133177932829, 20736286803363051714, 541224489038545084067, 14234799536039481373552, 376974819516101224941091

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=12 of A340811.

Cf. A054581.

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

A063688 Number of 2-trees rooted at a triangle.

Original entry on oeis.org

1, 1, 3, 10, 39, 164, 746, 3474, 16658, 81166, 401169, 2004517, 10110757, 51402250, 263133142, 1355126922, 7016115632, 36498130908, 190673015083, 999932115039, 5262094054524, 27779114013628, 147072756065567, 780722981065006

Offset: 1

Vladeta Jovovic, Aug 22 2001

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, Eq. (3.5.13).

Cf. A005750, A058870, A058866, A054581.

A322754 Number of unlabeled 7-trees on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 5, 15, 64, 342, 2344, 19137, 181098, 1922215, 22472875, 284556458, 3849828695, 54974808527, 819865209740, 12655913153775, 200748351368185, 3253193955012557, 53619437319817482, 895778170144927928, 15129118461773051724

Offset: 1

N. J. A. Sloane, Dec 26 2018

nonn

A k-tree is recursively defined as follows: K_k is a k-tree and any k-tree on n+1 vertices is obtained by joining a vertex to a k-clique in a k-tree on n vertices.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 328.

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45.

Column k=7 of A370770.

Cf. A054581 (unlabeled 2-trees), A078792 (unlabeled 3-trees), A078793 (unlabeled 4-trees), A201702 (unlabeled 5-trees), A202037 (unlabeled 6-trees).

A063682 Number of 2-trees rooted at an asymmetric end-edge.

Original entry on oeis.org

0, 1, 3, 13, 52, 229, 1024, 4749, 22454, 108299, 529896, 2625903, 13148268, 66428383, 338197609, 1733399552, 8936746984, 46315049443, 241146306914, 1260805087503, 6616786338256, 34843842405490, 184057255372576

Offset: 1

Vladeta Jovovic, Aug 22 2001

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 75, Eq. (3.5.4).

Cf. A058870, A058866, A054581.

A063689 Number of 2-trees rooted at a triangle with 3 similar edges.

Original entry on oeis.org

1, 1, 2, 6, 21, 83, 356, 1599, 7434, 35381, 171508, 843419, 4197179, 21094355, 106915928, 545859112, 2804656069, 14491370996, 75248398034, 392476363133, 2055245992376, 10801442696736, 56953957110855, 301207378815752, 1597342159296786, 8492297139795170

Offset: 1

Vladeta Jovovic, Aug 22 2001

nonn

			Sequence really begins 1, 0, 0, 1, 0, 0, 2, 0, 0, 6, 0, 0, 21, 0, 0, 83, 0, 0, 356, ... but only nonzero trisection is shown.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, Eq. (3.5.17).

Cf. A005750, A058870, A058866, A054581.

a(n) = A058870(n) + A063687(n) with A058870(0)=0. - Sean A. Irvine, May 07 2023

More terms from Sean A. Irvine, May 07 2023

A063692 Number of 2-trees rooted at a triangle with two similar edges.

Original entry on oeis.org

1, 2, 2, 7, 11, 25, 43, 106, 180, 453, 797, 2023, 3632, 9328, 16960, 44036, 80989, 211815, 393098, 1034958, 1934813, 5121356, 9633260, 25615432, 48433926, 129289382, 245554773, 657691061, 1253974468, 3368475942, 6444250241

Offset: 2

Vladeta Jovovic, Aug 23 2001

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, Eq. (3.5.16).

Cf. A005750, A058870, A058866, A054581.

A173249 Partial sums of A000272.

Original entry on oeis.org

1, 2, 3, 6, 22, 147, 1443, 18250, 280394, 5063363, 105063363, 2463011054, 64380375278, 1856540769315, 58550453144611, 2004745521503986, 74062339559431922, 2936485391069247715, 124376016487663499491

Offset: 0

Jonathan Vos Post, Feb 13 2010

Partial sums of number of trees on n labeled nodes. The subsequence of primes in this sequence begin: 2, 58550453144611, no more through a(30).

			a(19) = 1 + 1 + 1 + 3 + 16 + 125 + 1296 + 16807 + 262144 + 4782969 + 100000000 + 2357947691 + 61917364224 + 1792160394037 + 56693912375296 + 1946195068359375 + 72057594037927936 + 2862423051509815793 + 121439531096594251776 + 5480386857784802185939.

Cf. A000055, A000169, A000312, A007778, A007830, A008785-A008791, A033842, A000272, A036361, A036362, A036506, A000055, A054581, A097170

a(n) = SUM[i=0..n] A000272(i) = SUM[i=0..n] i^(i-2).

A224917 Stable k-tree numbers.

Original entry on oeis.org

1, 1, 1, 2, 5, 15, 64, 342, 2344, 19137, 181204, 1927017, 22652805, 290392448, 4022276630, 59749492128, 946174967813, 15892939156209

Offset: 0

Ira M. Gessel, Apr 19 2013

a(n) is the number of unlabeled k-trees with n+k vertices for all k >= n-2.

A k-tree is recursively defined as follows: The complete graph K_k is a k-tree and a k-tree on n+1 vertices is obtained by joining a vertex to a k-clique in a k-tree on n vertices.

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
A. Gainer-Dewar, Γ-species and the enumeration of k-trees, Electron. J. Combin. 19, no. 4, (2012), P45.
I. M. Gessel and A. Gainer-Dewar, Counting unlabeled k-trees, arXiv:1309.1429 [math.CO], 2013-2014.
I. M. Gessel and A. Gainer-Dewar, Counting unlabeled k-trees, J. Combin. Theory Ser. A 126 (2014), 177-193.

Cf. A000055 (unlabeled trees), A054581 (unlabeled 2-trees), A078792 (unlabeled 3-trees), A078793 (unlabeled 4-trees), A201702 (unlabeled 5-trees), A202037 (unlabeled 6-trees).

cp's OEIS Frontend

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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A094656 Number of unlabeled 12-gonal 2-trees with n 12-gons.

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A063688 Number of 2-trees rooted at a triangle.

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A322754 Number of unlabeled 7-trees on n nodes.

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A063682 Number of 2-trees rooted at an asymmetric end-edge.

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A063689 Number of 2-trees rooted at a triangle with 3 similar edges.

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A063692 Number of 2-trees rooted at a triangle with two similar edges.

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A173249 Partial sums of A000272.

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A224917 Stable k-tree numbers.

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