A006081 -id:A006081 - 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David Feldman, Counting plane trees, Unpublished manuscript, 1992.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

A006080 Number of rooted projective plane trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 56, 155, 469, 1480, 4882, 16545, 57384, 202060, 720526, 2593494, 9408469, 34350507, 126109784, 465200333, 1723346074, 6408356210, 23911272090, 89495909409, 335916761128, 1264114452996, 4768464309416, 18027250459483, 68291947831046, 259200707489634

Offset: 1

N. J. A. Sloane

Number of rooted planar trees that can be turned over.

Also bracelets (or necklaces) with n-1 black beads and n-1 white beads such that the beads switch colors when bracelet is turned over.

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

T. D. Noe, Table of n, a(n) for n=1..200
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
Index entries for sequences related to rooted trees
Index entries for sequences related to bracelets

Cf. A006079, A006081, A006082.

a[n_] := Sum[ EulerPhi[(n-1)/k]*(Binomial[2*k, k]/(2*(n-1))), {k, Divisors[n-1]}]/2 + 2^(n-3); a[1] = 1; Table[a[n], {n, 1, 27}] (* From Jean-François Alcover, Apr 11 2012, from formula *)

C(n, k)=binomial(n, k);
A003239(n) = if(n<=0, n==0, sumdiv(n, d, eulerphi(n/d) * C(2*d, d)) / (2*n) );
a(n) = if ( n<=1, 1, A003239(n)/2 + 2^(n-2) );
/* Joerg Arndt, Jan 25 2013 */

from sympy import binomial as C, totient, divisors
def a003239(n): return 1 if n<2 else sum([totient(n//d)*C(2*d, d) for d in divisors(n)])/(2*n)
def a(n): return 1 if n<2 else a003239(n)/2 + 2**(n - 2) # Indranil Ghosh, Apr 24 2017

Stockmeyer gives g.f.

a(n) = A003239(n)/2 + 2^(n-2). (n>=2) (corrected, Joerg Arndt, Jan 25 2013)

More terms, formula and additional comments from Christian G. Bower, Dec 13 2001

A224850 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 1 element; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 5, 2, 12, 6, 1, 1, 3, 3, 5, 7, 17, 1, 1, 8, 3, 25, 11, 106, 44

Offset: 1

Christopher Hunt Gribble, Jul 22 2013

It appears that sequence T(2,k) consists of 2 interspersed Fibonacci sequences.

The diagonal T(n,n) is A006081. - M. F. Hasler, Jul 25 2013

			The triangle is:
n\k  1   2   3   4   5   6   7   8 ...
.
0    1   1   1   1   1   1   1   1 ...
1        1   1   1   1   1   1   1 ...
2            1   3   2   5   3   8 ...
3                1   2   2   3   3 ...
4                    3  12   5  25 ...
5                        6   7  11 ...
6                           17 106 ...
7                               44 ...
...
T(3,5) = 2 because there are 2 different tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will only transform each tiling into itself.  Group D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
The tilings are:
._________.    ._________.
|_|_|_|_|_|    |_|     |_|
|_|_|_|_|_|    |_|     |_|
|_|_|_|_|_|    |_|_____|_|

Christopher Hunt Gribble, C++ program

Cf. A219924, A224697, A227690.

T(n,k) + A224861(n,k) + A224867(n,k) = A227690(n,k).

1*T(n,k) + 2*A224861(n,k) + 4*A224867(n,k) = A219924(n,k).

A384967 Number of unsensed simple planar maps with n vertices and 2 faces.

Original entry on oeis.org

0, 0, 1, 2, 7, 22, 76, 271, 1001, 3765, 14381, 55450, 214880, 835663, 3255652, 12698352, 49559793, 193513944, 755852101, 2953214386, 11541989533, 45123241746, 176465152051, 690340349398, 2701579878022, 10576116931462, 41418132927403, 162259989848094, 635899817853002, 2492993368347594

Offset: 1

Andrew Howroyd, Jun 15 2025

nonn

In other words, a(n) is the number of embeddings on the sphere of connected simple unicyclic planar graphs with n nodes.

Column 2 of A384963.
Also subdiagonal of A379430.
Cf. A001429, A006081 (cycle is loop), A380239 (not necessarily simple), A384966 (sensed version).

G1(n)={my(g=(1-sqrt(1-4*x^2 + O(x^(n+2))))/(2*x^2)); ((1 + x/(1-x-x^2*g)^2)^2/(1 - x^2*g^2) - 1)/2 + 1/(1 - x*g) - 1 - x*(g^2/(1 - x*g)^2 + g) - x^2*(g^4/(1 - x*g)^4 + 3*g^2)/2}
G2(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); sum(k=1, n, my(m=1+k%2); -(log(2 - c(k)) + log(1 - x^k*c(m*k)^(2/m)))*eulerphi(k)/k, O(x*x^n)) - x*(c(1)^2 + c(2)) - x^2*(c(1)^4 + 3*c(2)^2)/2}
seq(n)={Vec(G1(n)+G2(n), -n)/4}

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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A006080 Number of rooted projective plane trees with n nodes.

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A224850 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 1 element; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

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A384967 Number of unsensed simple planar maps with n vertices and 2 faces.

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