A302828 -id:A302828 - 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David Feldman, Counting plane trees, Unpublished manuscript, 1992.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

A303864 Array read by antidiagonals: T(n,k) = number of noncrossing path sets on k*n nodes up to rotation with each path having exactly k nodes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 6, 2, 1, 1, 4, 36, 38, 3, 1, 1, 10, 210, 960, 384, 6, 1, 1, 16, 1176, 18680, 35956, 4425, 14, 1, 1, 36, 6328, 313664, 2280910, 1588192, 57976, 34, 1, 1, 64, 32896, 4683168, 111925464, 323840016, 77381016, 807318, 95, 1

Offset: 0

Andrew Howroyd, May 01 2018

			Array begins:
=======================================================
n\k| 1  2     3        4           5              6
---+---------------------------------------------------
0  | 1  1     1        1           1              1 ...
1  | 1  1     1        3           4             10 ...
2  | 1  1     6       36         210           1176 ...
3  | 1  2    38      960       18680         313664 ...
4  | 1  3   384    35956     2280910      111925464 ...
5  | 1  6  4425  1588192   323840016    46552781760 ...
6  | 1 14 57976 77381016 50668922540 21346459738384 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Columns 2..4 are A002995(n+1), A303865, A303866.
Row n=1 is A051437(k-3).
Cf. A302828, A303844, A303869.
Cf. A295224 (polygon dissections), A303694 (sets of cycles instead of paths).

nmax = 10; seq[n_, k_] := Module[{p, q, h}, p = 1 + InverseSeries[ x/(k*2^If[k == 1, 0, k - 3]*(1 + x)^k) + O[x]^n, x ]; h = p /. x -> x^2 + O[x]^n; q = x*D[p, x]/p; Integrate[((p - 1)/k + Sum[EulerPhi[d]*(q /. x -> x^d + O[x]^n), {d, 2, n}])/x, x] + If[OddQ[k], 0, 2^(k/2 - 2)*x*h^(k/2)] + 1];
Clear[col]; col[k_] := col[k] = CoefficientList[seq[nmax, k], x];
T[n_, k_] := col[k][[n + 1]];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 04 2018, after Andrew Howroyd *)

seq(n,k)={ \\ gives gf of k'th column
my(p=1 + serreverse( x/(k*2^if(k==1, 0, k-3)*(1 + x)^k) + O(x*x^n) ));
my(h=subst(p,x,x^2+O(x*x^n)), q=x*deriv(p)/p);
intformal( ((p-1)/k + sum(d=2,n,eulerphi(d)*subst(q,x,x^d+O(x*x^n))))/x) + if(k%2, 0, 2^(k/2-2)*x*h^(k/2)) + 1;
}
Mat(vector(6, k, Col(seq(7, k))))

A303330 a(n) is the number of noncrossing path sets on 3*n nodes up to rotation and reflection with each path having exactly 3 nodes.

Original entry on oeis.org

1, 1, 4, 22, 201, 2244, 29096, 404064, 5915838, 89918914, 1408072452, 22585364697, 369552118682, 6148989874890, 103788529623864, 1773645405777098, 30638842342771863, 534324445644633987, 9397210553851138484, 166518651072771792918, 2970743502941350443069

Offset: 0

J. Stauduhar, Apr 21 2018

nonn

Paths are constructed using noncrossing line segments between the vertices of a regular 3n-gon. Isolated vertices are not allowed.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Math StackExchange, Question from user Matan at math.stackexchange.com: Number of ways to connect sets of k dots in a perfect n-gon

Column k=3 of A302828.

Cf. A303839, A303865, A303867.

seq[n_] := Module[{p, h, q, c}, p = 1 + InverseSeries[x/(3*(1 + x)^3) + O[x]^n , x]; h = (p /. x -> x^2 + O[x]^n); q = x*D[p, x]/p; c = Integrate[((p - 1)/3 + Sum[EulerPhi[d]*(q /. x -> x^d + O[x]^n), {d, 2, n}])/x, x]; CoefficientList[1 + c + (1 + h + x^2*h^3 + x*2*h^2)/2, x]/2];
seq[30] (* Jean-François Alcover, Jul 05 2018, after Andrew Howroyd *)

seq(n)={
my(p=1 + serreverse( x/(3*(1 + x)^3) + O(x*x^n) ));
my(h=subst(p, x, x^2 + O(x*x^n)), q=x*deriv(p)/p);
my(c=intformal(((p-1)/3 + sum(d=2, n, eulerphi(d)*subst(q, x, x^d+O(x*x^n))))/x));
Vec(1 + c + (1 + h + x^2*h^3 + x*2*h^2)/2)/2} \\ Andrew Howroyd, Apr 29 2018

a(n) ~ 3^(4*n - 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 3)). - Vaclav Kotesovec, Jun 01 2022

Terms a(8) and beyond from Andrew Howroyd, Apr 29 2018

a(6) corrected by Andrew Howroyd, May 03 2018

A303867 Number of noncrossing path sets on 4*n nodes up to rotation and reflection with each path having exactly 4 nodes.

Original entry on oeis.org

1, 2, 21, 494, 18086, 794696, 38695548, 2015556488, 110292751866, 6267709291736, 367003473639464, 22018423100856184, 1347856204419978236, 83918845269760695536, 5300972002005297517812, 339058084617031980524000, 21924124400037221008705338, 1431303944222490626674244672

Offset: 0

Andrew Howroyd, May 01 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
Math StackExchange, Question from user Matan at math.stackexchange.com: Number of ways to connect sets of k dots in a perfect n-gon

Column 4 of A302828.

Cf. A303330, A303839, A303866.

a(n) ~ 2^(11*n - 5/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jun 01 2022

A303868 Triangle read by rows: T(n,k) = number of noncrossing path sets on n nodes up to rotation and reflection with k paths and isolated vertices allowed.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 3, 7, 6, 2, 1, 6, 20, 23, 11, 3, 1, 10, 50, 80, 51, 17, 3, 1, 20, 136, 285, 252, 109, 26, 4, 1, 36, 346, 966, 1119, 652, 200, 36, 4, 1, 72, 901, 3188, 4782, 3623, 1502, 352, 50, 5, 1, 136, 2264, 10133, 19116, 18489, 9949, 3120, 570, 65, 5, 1

Offset: 1

Andrew Howroyd, May 01 2018

			Triangle begins:
   1;
   1,   1;
   1,   1,    1;
   2,   3,    2,    1;
   3,   7,    6,    2,    1;
   6,  20,   23,   11,    3,    1;
  10,  50,   80,   51,   17,    3,   1;
  20, 136,  285,  252,  109,   26,   4,  1;
  36, 346,  966, 1119,  652,  200,  36,  4, 1;
  72, 901, 3188, 4782, 3623, 1502, 352, 50, 5, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A303835.
Column 1 is A005418(n-2).
Cf. A302828, A303869.

\\ See A303731 for NCPathSetsModDihedral
{ my(rows=Vec(NCPathSetsModDihedral(vector(10, k, y))-1));
  for(n=1, #rows, for(k=1, n, print1(polcoeff(rows[n],k), ", ")); print;) }

cp's OEIS Frontend

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

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A303864 Array read by antidiagonals: T(n,k) = number of noncrossing path sets on k*n nodes up to rotation with each path having exactly k nodes.

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A303330 a(n) is the number of noncrossing path sets on 3*n nodes up to rotation and reflection with each path having exactly 3 nodes.

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Mathematica

PARI

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A303867 Number of noncrossing path sets on 4*n nodes up to rotation and reflection with each path having exactly 4 nodes.

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Formula

A303868 Triangle read by rows: T(n,k) = number of noncrossing path sets on n nodes up to rotation and reflection with k paths and isolated vertices allowed.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI