A358579 -id:A358579 - cp's OEIS frontend

A000891 a(n) = (2n)!(2n+1)! / (n! (n+1)!)^2.

1, 3, 20, 175, 1764, 19404, 226512, 2760615, 34763300, 449141836, 5924217936, 79483257308, 1081724803600, 14901311070000, 207426250094400, 2913690606794775, 41255439318353700, 588272005095043500

Offset: 0

N. J. A. Sloane

nonn

Number of parallelogram polyominoes having n+1 columns and n+1 rows. - Emeric Deutsch, May 21 2003
Number of tilings of an  hexagon.
a(n) is the number of non-crossing partitions of [2n+1] into n+1 blocks. For example, a[1] counts 13-2, 1-23, 12-3. - David Callan, Jul 25 2005
The number of returning walks of length 2n on the upper half of a square lattice, since a(n) = Sum_{k=0..2n} binomial(2n,k)*A126120(k)*A126869(n-k). - Andrew V. Sutherland, Mar 24 2008
For sequences counting walks in the upper half-plane starting from the origin and finishing at the lattice points (0,m) see A145600 (m = 1), A145601 (m = 2), A145602 (m = 3) and A145603 (m = 4). - Peter Bala, Oct 14 2008
The number of proper mergings of two n-chains. - Henri Mühle, Aug 17 2012
a(n) is number of pairs of non-intersecting lattice paths from (0,0) to (n+1,n+1) using (1,0) and (0,1) as steps. Here, non-intersecting means two paths do not share a vertex except the origin and the destination. For example, a(1) = 3 because we have three such pairs from (0,0) to (2,2): {NNEE,EENN}, {NNEE,ENEN}, {NENE,EENN}. - Ran Pan, Oct 01 2015
Also the number of ordered rooted trees with 2(n+1) nodes and n+1 leaves, i.e., half of the nodes are leaves. These trees are ranked by A358579. The unordered version is A185650. - Gus Wiseman, Nov 27 2022
The number of secondary GL(2) invariants constructed from n+1 two component vectors. This number was evaluated by using the Molien-Weyl formula to compute the Hilbert series of the ring of invariants. - Jaco van Zyl, Jun 30 2025

			G.f. = 1 + 3*x + 20*x^2 + 175*x^3 + 1764*x^4 + 19404*x^5 + ...
From _Gus Wiseman_, Nov 27 2022: (Start)
The a(2) = 20 ordered rooted trees with 6 nodes and 3 leaves:
  (((o)oo))  (((o)o)o)  (((o))oo)
  (((oo)o))  (((oo))o)  ((o)(o)o)
  (((ooo)))  ((o)(oo))  ((o)o(o))
  ((o(o)o))  ((o(o))o)  (o((o))o)
  ((o(oo)))  ((oo)(o))  (o(o)(o))
  ((oo(o)))  (o((o)o))  (oo((o)))
             (o((oo)))
             (o(o(o)))
(End)

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 94.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
Elena Barcucci, Andrea Frosini and Simone Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011), Article 11.4.5.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, arXiv:0804.2930 [math.CO], 2008.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, Discrete Math., 309 (2009), 2834-2838.
Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.
Ivan Marin and Emmanuel Wagner, A cubic defining algebra for the Links-Gould polynomial. arXiv preprint arXiv:1203.5981 [math.GT], 2012. - From _N. J. A. Sloane_, Sep 21 2012
Henri Mühle, Counting Proper Mergings of Chains and Antichains, arXiv:1206.3922 [math.CO], 2012.
Guoce Xin, Determinant formulas relating to tableaux of bounded height, Adv. Appl. Math. 45 (2010) 197-211.

Cf. A000356, A010370, A038535.
Cf. A145600, A145601, A145602, A145603. - Peter Bala, Oct 14 2008
Cf. A000142, A000894, A248045.
Equals half of A267981.
Counts the trees ranked by A358579.
A001263 counts ordered rooted trees by nodes and leaves.
A090181 counts ordered rooted trees by nodes and internals.
Cf. A000108, A065097, A080936, A185650, A358371, A358586, A358590.

a000891 n = a001263 (2 * n - 1) n  -- Reinhard Zumkeller, Oct 10 2013

[Factorial(2*n)*Factorial(2*n+1) / (Factorial(n) * Factorial(n+1))^2: n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

with(combstruct): bin := {B=Union(Z,Prod(B,B))} :seq(1/2*binomial(2*i,i)*(count([B,bin,unlabeled],size=i)), i=1..18) ; # Zerinvary Lajos, Jun 06 2007

a[ n_] := If[ n == -1, 0, Binomial[2 n + 1, n]^2 / (2 n + 1)]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (1 - Hypergeometric2F1[ -1/2, 1/2, 1, 16 x]) / (4 x), {x, 0, n}]; (* Michael Somos, May 28 2014 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ BesselI[0, 2 x] BesselI[1, 2 x] / x, {x, 0, 2 n}]]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticE[ 16 x] / (Pi/2)) / (4 x), {x, 0, n}]; (* Michael Somos, Sep 18 2016 *)
a[n_] := (2 n + 1) CatalanNumber[n]^2;
Array[a, 20, 0] (* Peter Luschny, Mar 03 2020 *)

{a(n) = binomial(2*n+1, n)^2 / (2*n + 1)}; /* Michael Somos, Jun 22 2005 */

a(n) = matdet(matrix(n, n, i, j, binomial(n+j+1,i+1))) \\ Hugo Pfoertner, Oct 22 2022

-4*a(n) = A010370(n+1).
G.f.: (1 - E(16*x)/(Pi/2))/(4*x) where E() is the elliptic integral of the second kind. [edited by Olivier Gérard, Feb 16 2011]
G.f.: 3F2(1, 1/2, 3/2; 2,2; 16*x)= (1 - 2F1(-1/2, 1/2; 1; 16*x)) / (4*x). - Olivier Gérard, Feb 16 2011
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2*x) * BesselI(1, 2*x) / x. - Michael Somos, Jun 22 2005
a(n) = A001700(n)*A000108(n) = (1/2)*A000984(n+1)*A000108(n). - Zerinvary Lajos, Jun 06 2007
For n > 0, a(n) = (n+2)*A000356(n) starting (1, 5, 35, 294, ...). - Gary W. Adamson, Apr 08 2011
a(n) = A001263(2*n+1,n+1) = binomial(2*n+1,n+1)*binomial(2*n+1,n)/(2*n+1) (central members of odd numbered rows of Narayana triangle).
G.f.: If G_N(x) = 1 + Sum_{k=1..N} ((2*k)!*(2*k+1)!*x^k)/(k!*(k+1)!)^2, G_N(x) = 1 + 12*x/(G(0) - 12*x); G(k) = 16*x*k^2 + 32*x*k + k^2 + 4*k + 12*x + 4 - 4*x*(2*k+3)*(2*k+5)*(k+2)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
D-finite with recurrence (n+1)^2*a(n) - 4*(2*n-1)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
a(n) = A005558(2n). - Mark van Hoeij, Aug 20 2014
a(n) = A000894(n) / (n+1) = A248045(n+1) / A000142(n+1). - Reinhard Zumkeller, Sep 30 2014
From Ilya Gutkovskiy, Feb 01 2017: (Start)
E.g.f.: 2F2(1/2,3/2; 2,2; 16*x).
a(n) ~ 2^(4*n+1)/(Pi*n^2). (End)
a(n) = A005408(n)*(A000108(n))^2. - Ivan N. Ianakiev, Nov 13 2019
a(n) = det(M(n)) where M(n) is the n X n matrix with m(i,j) = binomial(n+j+1,i+1). - Benoit Cloitre, Oct 22 2022
a(n) = Integral_{x=0..16} x^n*W(x) dx, where W(x) = (16*EllipticE(1 - x/16) - x*EllipticK(1 - x/16))/(8*Pi^2*sqrt(x)), n=>0. W(x) diverges at x=0, monotonically decreases for x>0, and vanishes at x=16. EllipticE and EllipticK are elliptic functions. This integral representation as n-th moment of a positive function W(x) on the interval [0, 16] is unique. - Karol A. Penson, Dec 20 2023

More terms from Andrew V. Sutherland, Mar 24 2008

A185650 a(n) is the number of rooted trees with 2n vertices n of whom are leaves.

Original entry on oeis.org

1, 2, 8, 39, 214, 1268, 7949, 51901, 349703, 2415348, 17020341, 121939535, 885841162, 6511874216, 48359860685, 362343773669, 2736184763500, 20805175635077, 159174733727167, 1224557214545788, 9467861087020239, 73534456468877012, 573484090227222260

Offset: 1

Stepan Orevkov, Aug 29 2013

nonn

			From _Gus Wiseman_, Nov 27 2022: (Start)
The a(1) = 1 through a(3) = 8 rooted trees:
  (o)  ((oo))  (((ooo)))
       (o(o))  ((o)(oo))
               ((o(oo)))
               ((oo(o)))
               (o((oo)))
               (o(o)(o))
               (o(o(o)))
               (oo((o)))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
V. M. Kharlamov and S. Yu. Orevkov, The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves, arXiv:math/0301245 [math.AG], 2003; J. of Combinatorial Theory, Ser. A, 105 (2004), 127-142.
Index entries for sequences related to rooted trees

The ordered version is A000891, ranked by A358579.
This is the central column of A055277.
These trees are ranked by A358578.
For height = internals we have A358587.
Square trees are counted by A358589.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internals, ordered A090181.
Cf. A034781, A109129, A358580, A358581-A358584, A358591.

terms = 23;
m = 2 terms;
T[, ] = 0;
Do[T[x_, z_] = z x - x + x Exp[Sum[Series[1/k T[x^k, z^k], {x, 0, j}, {z, 0, j}], {k, 1, j}]] // Normal, {j, 1, m}];
cc = CoefficientList[#, z]& /@ CoefficientList[T[x, z] , x];
Table[cc[[2n+1, n+1]], {n, 1, terms}] (* Jean-François Alcover, Sep 14 2018 *)
art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{-2}]==n/2&]],{n,2,10,2}] (* Gus Wiseman, Nov 27 2022 *)

\\ here R is A055277 as vector of polynomials
R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
{my(A=R(2*30)); vector(#A\2, k, polcoeff(A[2*k],k))} \\ Andrew Howroyd, May 21 2018

Terms a(20) and beyond from Andrew Howroyd, May 21 2018

A358577 Matula-Goebel numbers of "square" rooted trees, i.e., whose height equals their number of leaves.

Original entry on oeis.org

1, 4, 12, 14, 18, 19, 21, 27, 40, 52, 60, 68, 70, 74, 78, 86, 89, 90, 91, 92, 95, 100, 102, 105, 107, 111, 117, 119, 122, 129, 130, 134, 135, 138, 146, 150, 151, 153, 161, 163, 169, 170, 175, 176, 181, 183, 185, 195, 201, 206, 207, 215, 219, 221, 225, 227, 230

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The terms together with their corresponding rooted trees begin:
   1: o
   4: (oo)
  12: (oo(o))
  14: (o(oo))
  18: (o(o)(o))
  19: ((ooo))
  21: ((o)(oo))
  27: ((o)(o)(o))
  40: (ooo((o)))
  52: (oo(o(o)))
  60: (oo(o)((o)))
  68: (oo((oo)))
  70: (o((o))(oo))
  74: (o(oo(o)))
  78: (o(o)(o(o)))
  86: (o(o(oo)))
  89: ((ooo(o)))
  90: (o(o)(o)((o)))

Gus Wiseman, The first 64 square rooted trees.

Internals instead of leaves: A358576, counted by A358587, ordered A358588.
Internals instead of height: A358578, counted by A185650, ordered A358579.
These trees are counted by A358589, ordered A358590.
A000081 counts rooted trees, ordered A000108.
A034781 counts trees by nodes and height.
A055277 counts trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358379, A358580, A358581-A358586, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],{},{0,Infinity}]==Depth[MGTree[#]]-1&]

A358552(a(n)) = A109129(a(n)).

A358578 Matula-Goebel numbers of rooted trees whose number of leaves equals their number of internal (non-leaf) nodes.

Original entry on oeis.org

2, 6, 7, 18, 20, 21, 26, 34, 37, 43, 54, 60, 63, 67, 70, 78, 88, 91, 92, 95, 102, 111, 116, 119, 122, 129, 142, 146, 151, 162, 164, 173, 180, 181, 189, 200, 201, 202, 210, 227, 234, 236, 239, 245, 260, 264, 269, 273, 276, 278, 285, 306, 308, 314, 322, 333, 337

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

			The terms together with their corresponding rooted trees begin:
   2: (o)
   6: (o(o))
   7: ((oo))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  26: (o(o(o)))
  34: (o((oo)))
  37: ((oo(o)))
  43: ((o(oo)))
  54: (o(o)(o)(o))
  60: (oo(o)((o)))
  63: ((o)(o)(oo))
  67: (((ooo)))
  70: (o((o))(oo))
  78: (o(o)(o(o)))
  88: (ooo(((o))))
  91: ((oo)(o(o)))

Gus Wiseman, The first 64 rooted trees whose number of leaves equals their number of internal nodes.

These trees are counted by A185650, ordered A358579.
Height instead of leaves: A358576, counted by A358587, ordered A358588.
Height instead of internals: A358577, counted by A358589, ordered A358590.
Positions of 0's in A358580.
A000081 counts rooted trees, ordered A000108.
A034781 counts trees by nodes and height.
A055277 counts trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358371, A358581-A358586, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],{},{0,Infinity}]==Count[MGTree[#],[_],{0,Infinity}]&]

A342507(a(n)) = A109129(a(n)).

A358580 Difference between the number of leaves and the number of internal (non-leaf) nodes in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 0, -1, 1, -2, 0, 0, 2, -1, -1, -3, 1, -1, 1, -2, 3, -1, 0, 1, 0, 0, -2, -2, 2, -3, 0, -1, 2, -2, -1, -4, 4, -3, 0, -1, 1, 0, 2, -1, 1, -2, 1, 0, -1, -2, -1, -3, 3, 1, -2, -1, 1, 2, 0, -4, 3, 1, -1, -2, 0, -1, -3, 0, 5, -2, -2, 0, 1, -2, 0, -1, 2, -1, 1, -3

Offset: 1

Gus Wiseman, Nov 25 2022

sign

			The Matula-Goebel number of ((ooo(o))) is 89, and it has 4 leaves and 3 internal nodes, so a(89) = 1.

Zeros are A358578, counted by A185650 (ordered A358579).
Positions of positive terms are counted by A358581, negative A358582.
Positions of nonnegative terms are counted by A358583, nonpositive A358584.
A000081 counts rooted trees, ordered A000108.
A034781 counts trees by nodes and height.
A055277 counts trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358371, A358576, A358577, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],{},{0,Infinity}]-Count[MGTree[n],[_],{0,Infinity}],{n,100}]

a(n) = A109129(n) - A342507(n).

A358590 Number of square ordered rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 6, 5, 36, 84, 309, 890, 3163, 9835, 32979, 108252, 360696, 1192410, 3984552, 13276769, 44371368, 148402665, 497072593, 1665557619, 5586863093, 18750662066, 62968243731, 211565969511, 711187790166, 2391640404772, 8045964959333, 27077856222546

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

We say that a tree is square if it has the same height as number of leaves.

			The a(1) = 1 through a(6) = 5 ordered trees:
  o  .  (oo)  .  ((o)oo)  ((o)(o)o)
                 ((oo)o)  ((o)(oo))
                 ((ooo))  ((o)o(o))
                 (o(o)o)  ((oo)(o))
                 (o(oo))  (o(o)(o))
                 (oo(o))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For internals instead of height we have A000891, unordered A185650 aerated.
For internals instead of leaves we have A358588, unordered A358587.
The unordered version is A358589, ranked by A358577.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
Cf. A065097, A109129, A358371, A358552, A358579, A358586.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,{},{0,Infinity}]==Depth[#]-1&]],{n,1,10}]

\\ R(n,f) enumerates trees by height(h), nodes(x) and leaves(y).
R(n,f) = {my(A=O(x*x^n), Z=0); for(h=1, n, my(p = A); A = x*(y - 1  + 1/(1 - A + O(x^n))); Z += f(h, A-p)); Z}
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(16) and beyond from Andrew Howroyd, Jan 01 2023

A358586 Number of ordered rooted trees with n nodes, at least half of which are leaves.

Original entry on oeis.org

1, 1, 1, 4, 7, 31, 66, 302, 715, 3313, 8398, 39095, 104006, 484706, 1337220, 6227730, 17678835, 82204045, 238819350, 1108202513, 3282060210, 15195242478, 45741281820, 211271435479, 644952073662, 2971835602526, 9183676536076, 42217430993002, 131873975875180, 604834233372884

Offset: 1

Gus Wiseman, Nov 24 2022

nonn

			The a(1) = 1 through a(5) = 7 ordered trees:
  o  (o)  (oo)  (ooo)   (oooo)
                ((o)o)  ((o)oo)
                ((oo))  ((oo)o)
                (o(o))  ((ooo))
                        (o(o)o)
                        (o(oo))
                        (oo(o))

Andrew Howroyd, Table of n, a(n) for n = 1..1000

For equality we have A000891, unordered A185650.
Odd-indexed terms appear to be A065097.
The unordered version is A358583.
The opposite is the same, unordered A358584.
The strict case is A358585, unordered A358581.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
A358590 counts square ordered trees, unordered A358589 (ranked by A358577).
Cf. A109129, A342507, A358371, A358579, A358580, A358582, A358584, A358588.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,{},{0,Infinity}]>=Count[#,[_],{0,Infinity}]&]],{n,1,10}]

a(n) = if(n==1, 1, n--; (binomial(2*n,n)/(n+1) + if(n%2, binomial(n, (n-1)/2)^2 / n))/2) \\ Andrew Howroyd, Jan 13 2024

From Andrew Howroyd, Jan 13 2024: (Start)
a(n) = Sum_{k=1..floor(n/2)} A001263(n-1, k) for n >= 2.
a(2*n) = (A000108(2*n-1) + A000891(n-1))/2 for n >= 1;
a(2*n+1) = A000108(2*n)/2 for n >= 1. (End)

a(16) onwards from Andrew Howroyd, Jan 13 2024

A358581 Number of rooted trees with n nodes, most of which are leaves.

Original entry on oeis.org

1, 0, 1, 1, 4, 5, 20, 28, 110, 169, 663, 1078, 4217, 7169, 27979, 49191, 191440, 345771, 1341974, 2477719, 9589567, 18034670, 69612556, 132984290, 511987473, 991391707, 3807503552, 7460270591, 28585315026, 56595367747, 216381073935, 432396092153

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(1) = 1 through a(7) = 20 trees:
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)
                     ((ooo))  ((oooo))  ((ooooo))
                     (o(oo))  (o(ooo))  (o(oooo))
                     (oo(o))  (oo(oo))  (oo(ooo))
                              (ooo(o))  (ooo(oo))
                                        (oooo(o))
                                        (((oooo)))
                                        ((o)(ooo))
                                        ((o(ooo)))
                                        ((oo)(oo))
                                        ((oo(oo)))
                                        ((ooo(o)))
                                        (o((ooo)))
                                        (o(o)(oo))
                                        (o(o(oo)))
                                        (o(oo(o)))
                                        (oo((oo)))
                                        (oo(o)(o))
                                        (oo(o(o)))
                                        (ooo((o)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For equality we have A185650 aerated, ranked by A358578.
The opposite version is A358582, non-strict A358584.
The non-strict version is A358583.
The ordered version is A358585, odd-indexed terms A065097.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
A358589 counts square trees, ranked by A358577, ordered A358590.
Cf. A000891, A109129, A342507, A358579, A358580, A358586, A358591.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}]>Count[#,[_],{0,Infinity}]&]],{n,0,10}]

\\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[n\2+1..#u]))} \\ Andrew Howroyd, Dec 31 2022

A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=floor(n/2)+1..n} A055277(n, k). - Andrew Howroyd, Dec 31 2022

Terms a(19) and beyond from Andrew Howroyd, Dec 31 2022

A358592 Matula-Goebel numbers of rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

Original entry on oeis.org

18, 21, 60, 70, 78, 91, 92, 95, 102, 111, 119, 122, 129, 146, 151, 181, 201, 227, 264, 269, 308, 348, 376, 406, 418, 426, 452, 492, 497, 519, 551, 562, 574, 583, 596, 606, 659, 664, 668, 698, 707, 708, 717, 779, 794, 796, 809, 826, 834, 911, 932, 934, 942, 958

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

			The terms together with their corresponding rooted trees begin:
   18: (o(o)(o))
   21: ((o)(oo))
   60: (oo(o)((o)))
   70: (o((o))(oo))
   78: (o(o)(o(o)))
   91: ((oo)(o(o)))
   92: (oo((o)(o)))
   95: (((o))(ooo))
  102: (o(o)((oo)))
  111: ((o)(oo(o)))
  119: ((oo)((oo)))
  122: (o(o(o)(o)))
  129: ((o)(o(oo)))
  146: (o((o)(oo)))
  151: ((oo(o)(o)))
  181: ((o(o)(oo)))
  201: ((o)((ooo)))
  227: (((oo)(oo)))

Gus Wiseman, The first 64 ordered trees whose height, number of leaves, and number of internal nodes are all equal.

Any number of leaves: A358576, counted by A358587 (ordered A358588).
Any number of internals: A358577, counted by A358589, ordered A358590.
Any height: A358578, ordered A358579, counted by A185650.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358580, A358581-A358586.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],[_],{0,Infinity}]==Count[MGTree[#],{},{0,Infinity}]==Depth[MGTree[#]]-1&]

A358552(a(n)) = A342507(a(n)) = A109129(a(n)).

A358588 Number of n-node ordered rooted trees of height equal to the number of internal (non-leaf) nodes.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 41, 171, 633, 2171, 7070, 22195, 67830, 203130, 598806, 1743258, 5023711, 14356226, 40737383, 114904941, 322432215, 900707165, 2506181060, 6948996085, 19207795836, 52944197508, 145567226556, 399314965956, 1093107693133, 2986640695436

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

			The a(5) = 1 and a(6) = 8 ordered trees:
  ((o)(o))  ((o)(o)o)
            ((o)(oo))
            ((o)o(o))
            ((oo)(o))
            (o(o)(o))
            (((o))(o))
            (((o)(o)))
            ((o)((o)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For leaves instead of height we have A000891, unordered A185650 aerated.
The unordered version is A358587, ranked by A358576.
For leaves instead of internal nodes we have A358590, unordered A358589.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
Cf. A065097, A342507, A358552, A358371, A358578, A358579, A358586, A358591.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,[_],{0,Infinity}]==Depth[#]-1&]],{n,1,10}]

\\ Needs R(n,f) defined in A358590.
seq(n) = {Vec(R(n, (h,p)->polcoef(subst(p, x, x/y), -h, y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Conjectures from Chai Wah Wu, Apr 14 2024: (Start)
a(n) = 9*a(n-1) - 32*a(n-2) + 58*a(n-3) - 58*a(n-4) + 32*a(n-5) - 9*a(n-6) + a(n-7) for n > 7.
G.f.: x^5*(-x^2 + x - 1)/((x - 1)^3*(x^2 - 3*x + 1)^2). (End)

Terms a(16) and beyond from Andrew Howroyd, Jan 01 2023

cp's OEIS Frontend

A000891 a(n) = (2*n)!*(2*n+1)! / (n! * (n+1)!)^2.

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A185650 a(n) is the number of rooted trees with 2n vertices n of whom are leaves.

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A358577 Matula-Goebel numbers of "square" rooted trees, i.e., whose height equals their number of leaves.

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A358578 Matula-Goebel numbers of rooted trees whose number of leaves equals their number of internal (non-leaf) nodes.

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A358580 Difference between the number of leaves and the number of internal (non-leaf) nodes in the rooted tree with Matula-Goebel number n.

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A358590 Number of square ordered rooted trees with n nodes.

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A358586 Number of ordered rooted trees with n nodes, at least half of which are leaves.

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A000891 a(n) = (2n)!(2n+1)! / (n! (n+1)!)^2.