A006080 -id:A006080 - cp's OEIS frontend

A006082 Number of labeled projective plane trees (or "flat" trees) with n nodes.

1, 1, 1, 2, 3, 6, 12, 27, 65, 175, 490, 1473, 4588, 14782, 48678, 163414, 555885, 1913334, 6646728, 23278989, 82100014, 291361744, 1039758962, 3729276257, 13437206032, 48620868106, 176611864312, 643834562075, 2354902813742, 8640039835974, 31791594259244

Offset: 1

N. J. A. Sloane

nonn

Also, the number of noncrossing partitions up to rotation and reflection composed of n-1 blocks of size 2. - Andrew Howroyd, May 03 2018

R. W. Robinson, personal communication.
R. W. Robinson, Efficiency of power series operations for graph counting, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..500
David Feldman, Counting plane trees, Unpublished manuscript, 1992. (Annotated scanned copy)
Richard Kapolnai, Gabor Domokos, and Timea Szabo, Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes, Periodica Polytechnica Electrical Engineering, 56(1):11-10, 2012. Also arXiv:1206.1698 [cs.DM], 2012. See row 2 of Table 1.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360. [The sequence is mentioned on page 355, but because of a miscalculation it is given, incorrectly, as 1, 1, 1, 2, 3, 6, 12, 25. Thanks to _David Broadhurst_ for this information. - _N. J. A. Sloane_, Apr 06 2022]
Feng Rong, A note on the topological classification of complex polynomial differential equations with only centre singularities, Journal of Difference Equations and Applications, Volume 18, Issue 11, 2012. - From _N. J. A. Sloane_, Dec 27 2012
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]
Index entries for sequences related to trees

Column k=2 of A302828 and A303929.
Cf. A006079, A006080, A006081.
Cf. A002995 (noncrossing partitions into pairs up to rotations only), A126120, A001405, A185100.

u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #]&] + DivisorSum[GCD[n-1, k], EulerPhi[#]*Binomial[n*k/#, (n-1)/#]&])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];
T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[ Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;
a[n_] := T[n, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd and A303929 *)

\\ from David Broadhurst, Apr 06 2022, added by N. J. A. Sloane, Apr 06 2022
{A006082(n)=my(c(n)=binomial(2*n,n));
if(n<2,1,n--;(c(n)+if(n%2,2*n*(n+2),(n+1)^2)*c(n\2)
+(n+1)*sumdiv(n,d,if(d>2,eulerphi(d)*c(n/d))))/(4*n*(n+1)));}

a(n) = A006080(n) - A006081(n) + A126120(n-2). [Stockmeyer] [Corrected by Andrey Zabolotskiy, Apr 06 2021]
a(n) = (2 * A002995(n) + A126120(n-2) + A001405(n-1)) / 4 for n > 1. - Andrey Zabolotskiy, May 24 2018
There is a compact formula from David Broadhurst - see the Pari code - N. J. A. Sloane, Apr 06 2022.
a(n) ~ 2^(2*n-4) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Jun 01 2022

a(25) and a(26) from Robert W. Robinson, Oct 17 2006

a(27) and beyond from Andrew Howroyd, May 03 2018

A006079 Number of asymmetric planted projective plane trees with n+1 nodes; bracelets (reversible necklaces) with n black beads and n-1 white beads.

Original entry on oeis.org

1, 1, 0, 1, 4, 16, 56, 197, 680, 2368, 8272, 29162, 103544, 370592, 1335504, 4844205, 17672400, 64810240, 238795040, 883585406, 3281967832, 12232957152, 45740929104, 171529130786, 644950721584, 2430970600576, 9183671335776, 34766765428852, 131873955816880

Offset: 1

N. J. A. Sloane

"DHK[ n ](2n-1)" (bracelet, identity, unlabeled, n parts, evaluated at 2n) transform of 1,1,1,1,...

For n > 2, half the number of asymmetric Dyck (n-1)-paths. E.g., the two asymmetric 3-paths are UDUUDD and UUDDUD, so a(4) = 2/2 = 1. - David Scambler, Aug 23 2012

			For the asymmetric planted projective plane trees sequence we have a(5) = 4, a(6) = 16, a(7) = 56, ...

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

T. D. Noe, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2).
Z. M. Himwich and N. A. Rosenberg, Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees, arXiv:1901.04465 [qbio.PE], 2019; Adv. Appl. Math. 113 (2020), 101939. (Table 1 shows twice this sequence.)
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]
Index entries for sequences related to bracelets
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000029, A000031, A006080, A006081, A006082.

Equals half the difference of A000108 and A001405.

[1,1] cat [(Catalan(n) - Binomial(n, Floor(n/2)))/2: n in [2..40]]; // Vincenzo Librandi, Feb 16 2015

a[1] = a[2] = 1; a[n_] := (CatalanNumber[n-1] - Binomial[n-1, Floor[(n-1)/2]])/2; Table[ a[n], {n, 1, 26}] (* Jean-François Alcover, Mar 09 2012, after David Callan *)

Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = x/(1-x-x^2*c(x^2)) = g.f. for A001405. Then g.f. for the asymmetric planted projective plane trees sequence is (x*c(x)-d(x))/2 (the initial terms from this version are slightly different).

a(n+1) = (CatalanNumber(n) - binomial(n,floor(n/2)))/2 (for n>=3). - David Callan, Jul 14 2006

Alternative description and more terms from Christian G. Bower

A066315 Number of aperiodic bracelets (or necklaces) with n red and blue beads such that the beads switch colors when bracelet is turned over.

Original entry on oeis.org

1, 1, 3, 7, 20, 51, 154, 460, 1476, 4860, 16544, 57321, 202059, 720370, 2593470, 9408000, 34350506, 126108252, 465200332, 1723341185, 6408356052, 23911255544, 89495909408, 335916703284, 1264114452975, 4768464107355

Offset: 1

Christian G. Bower, Dec 13 2001

nonn

Index entries for sequences related to bracelets

a[n_] := Sum[MoebiusMu[n/d] A006080[d + 1], {d, Divisors[n]}];
Array[a, 26] (* Jean-François Alcover, Aug 28 2019 *)

Moebius transform of A006080(n+1).

A115123 Number of imprimitive (periodic) bracelets (or necklaces) with n red and blue beads such that the beads switch colors when bracelet is turned over.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 9, 4, 22, 1, 63, 1, 156, 24, 469, 1, 1532, 1, 4889, 158, 16546, 1, 57844, 21, 202061, 1480, 720533, 1, 2598406, 1, 9408469, 16548, 34350508, 175, 126167112, 1, 465200334, 202063, 1723346534, 1, 6409076632, 1, 23911272097, 2594970, 89495909410, 1

Offset: 1

Valery A. Liskovets, Jan 17 2006

a(p)=1 for prime p.

A006080[n_] := Sum[EulerPhi[(n - 1)/k] (Binomial[2k, k]/(2(n - 1))), {k, Divisors[n - 1]}]/2 + 2^(n - 3); A006080[1] = 1;
A066315[n_] := Sum[MoebiusMu[n/d] A006080[d + 1], {d, Divisors[n]}];
a[n_] := A006080[n + 1] - A066315[n];
Array[a, 60] (* Jean-François Alcover, Aug 28 2019 *)

a(n)=A006080(n+1) - A066315(n).

More terms from Jean-François Alcover, Aug 28 2019

cp's OEIS Frontend

A006082 Number of labeled projective plane trees (or "flat" trees) with n nodes.

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A006079 Number of asymmetric planted projective plane trees with n+1 nodes; bracelets (reversible necklaces) with n black beads and n-1 white beads.

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A066315 Number of aperiodic bracelets (or necklaces) with n red and blue beads such that the beads switch colors when bracelet is turned over.

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A115123 Number of imprimitive (periodic) bracelets (or necklaces) with n red and blue beads such that the beads switch colors when bracelet is turned over.

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Mathematica

Formula

Extensions