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A002995 Number of unlabeled planar trees (also called plane trees) with n nodes.
(Formerly M0805)
30
1, 1, 1, 1, 2, 3, 6, 14, 34, 95, 280, 854, 2694, 8714, 28640, 95640, 323396, 1105335, 3813798, 13269146, 46509358, 164107650, 582538732, 2079165208, 7457847082, 26873059986, 97239032056, 353218528324, 1287658723550, 4709785569184 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Noncrossing handshakes of 2(n-1) people (each using only one hand) on round table, up to rotations - Antti Karttunen, Sep 03 2000

Equivalently, the number of noncrossing partitions up to rotation composed of n-1 blocks of size 2. - Andrew Howroyd, May 04 2018

a(n), n>2, is also the number of oriented cacti on n-1 unlabeled nodes with all cutpoints of separation degree 2, i.e. ones shared only by two (cyclic) blocks. These are digraphs (without loops) that have a unique Eulerian tour. Such digraphs with labeled nodes are enumerated by A102693. - Valery A. Liskovets, Oct 19 2005

Labeled plane trees are counted by A006963. - David Callan, Aug 19 2014

This sequence is similar to A000055 but those trees are not embedded in a plane. - Michael Somos, Aug 19 2014

REFERENCES

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 285 (4.1.26), 291 (4.1.48)

A. Errera, De quelques problèmes d'analysis situs, Comptes Rend. Congr. Nat. Sci. Bruxelles, (1930), 106-110.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 67, (3.3.26).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.

A. Errera, Reviews of two articles on Analysis Situs, from Fortschritte [Annotated scanned copy]

D. Feldman, Counting plane trees, Unpublished manuscript, 1992. (Annotated scanned copy)

F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.

F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)

G. Labelle, Sur la symétrie et l'asymétrie des structures combinatoires, Theor. Comput. Sci. 117, No. 1-2, 3-22 (1993).

P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)

T. Mütze, Proof of the middle levels conjecture, arXiv preprint arXiv:1404.4442 [math.CO], 2014.

Torsten Mütze and Franziska Weber, Construction of 2-factors in the middle layer of the discrete cube, arXiv preprint arXiv:1111.2413 [math.CO], 2011.

T. Mütze and F. Weber, Construction of 2-factors in the middle layer of the discrete cube, Journal of Combinatorial Theory, Series A, 119(8) (2012), 1832-1855.

J. Sawada, Generating rooted and free plane trees, ACM Transactions on Algorithms, Vol. 2 No. 1 (2006), 1-13.

Seunghyun Seo and Heesung Shin, Two involutions on vertices of ordered trees, FPSAC'02 (2002). (see p_n).

Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233. See Table 1.

D. W. Walkup, The number of plane trees, Mathematika, vol. 19, No. 2 (1972), 200-204.

Index entries for sequences related to trees

FORMULA

G.f.: 1+B(x)+(C(x^2)-C(x)^2)/2 where B is g.f. of A003239 and C is g.f. of A000108(n-1).

a(n) = 1/(2*(n-1))*sum{d|(n-1)}(phi((n-1)/d)*binomial(2d, d)) - A000108(n-1)/2 + (if n is even) A000108(n/2-1)/2.

EXAMPLE

G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 14*x^7 + 34*x^8 + 95*x^9 + ...

a(7) = 14 = 11 + 3 because there are 11 trees with 7 nodes but three of them can be embedded in a plane in two ways. These three trees have degree sequences 4221111, 3321111, 3222111, where there are two trees with each degree sequence but in the first, the two nodes of degree two are adjacent, in the second, the two nodes of degree three are adjacent, and in the third, the node of degree three is adjacent to two nodes of degree two. - Michael Somos, Aug 19 2014

MAPLE

with (powseries): with (combstruct): n := 27: Order := n+2: sys := {C = Cycle(B), B = Union(Z, Prod(B, B))}: G003239 := (convert(gfseries(sys, unlabeled, x) [C(x)], polynom)) / x: G000108 := convert(taylor((1-sqrt(1-4*x)) / (2*x), x), polynom): G002995 := 1 + G003239 + (eval(G000108, x=x^2) - G000108^2)/2: A002995 := 1, 1, 1, seq(coeff(G002995, x^i), i=1..n); # Ulrich Schimke, Apr 05 2002

with(combinat): with(numtheory): m := 2: for p from 2 to 28 do s1 := 0: s2 := 0: for d from 1 to p do if p mod d = 0 then s1 := s1+phi(p/d)*binomial(m*d, d) fi: od: for d from 1 to p-1 do if gcd(m, p-1) mod d = 0 then s2 := s2+phi(d)*binomial((p*m)/d, (p-1)/d) fi: od: printf(`%d, `, (s1+s2)/(m*p)-binomial(m*p, p)/(p*(m-1)+1)) od : # Zerinvary Lajos, Dec 01 2006

MATHEMATICA

a[0] = a[1] = 1; a[n_] := (1/(2*(n-1)))*Sum[ EulerPhi[(n-1)/d]*Binomial[2*d, d], {d, Divisors[n-1]}] - CatalanNumber[n-1]/2 + If[ EvenQ[n], CatalanNumber[n/2-1]/2, 0]; Table[ a[n], {n, 0, 29}] (* Jean-François Alcover, Mar 07 2012, from formula *)

PROG

(PARI) catalan(n) = binomial(2*n, n)/(n+1);

a(n) = if (n<2, 1, n--; sumdiv(n, d, eulerphi(n/d)*binomial(2*d, d))/(2*n) - catalan(n)/2 + if ((n-1) % 2, 0, catalan((n-1)/2)/2)); \\ Michel Marcus, Jan 23 2016

CROSSREFS

Column k=2 of A303694 and A303864.

Cf. A000055, A000108, A002996, A003239, A005354, A057502, A061417, A064640.

Sequence in context: A010357 A190166 A238823 * A093467 A246640 A080408

Adjacent sequences:  A002992 A002993 A002994 * A002996 A002997 A002998

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms, formula from Christian G. Bower, Dec 15 1999

Name corrected ("labeled" --> "unlabeled") by David Callan, Aug 19 2014

STATUS

approved

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Last modified September 19 01:08 EDT 2018. Contains 315153 sequences. (Running on oeis4.)