A080936 -id:A080936 - cp's OEIS frontend

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

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

A080934 Square array read by antidiagonals of number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2n steps with all values less than or equal to k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 4, 1, 0, 1, 1, 2, 5, 8, 1, 0, 1, 1, 2, 5, 13, 16, 1, 0, 1, 1, 2, 5, 14, 34, 32, 1, 0, 1, 1, 2, 5, 14, 41, 89, 64, 1, 0, 1, 1, 2, 5, 14, 42, 122, 233, 128, 1, 0, 1, 1, 2, 5, 14, 42, 131, 365, 610, 256, 1, 0, 1, 1, 2, 5, 14, 42, 132, 417, 1094, 1597, 512, 1, 0

Offset: 0

Henry Bottomley, Feb 25 2003

Number of permutations in S_n avoiding both 132 and 123...k.
T(n,k) = number of rooted ordered trees on n nodes of depth <= k. Also, T(n,k) = number of {1,-1} sequences of length 2n summing to 0 with all partial sums are >=0 and <= k. Also, T(n,k) = number of closed walks of length 2n on a path of k nodes starting from (and ending at) a node of degree 1. - Mitch Harris, Mar 06 2004
Also T(n,k) = k-th coefficient in expansion of the rational function R(n), where R(1) = 1, R(n+1) = 1/(1-x*R(n)), which means also that lim(n->inf,R(n)) = g.f. of Catalan numbers (A000108) wherever it has real value (see Mansour article). - Clark Kimberling and Ralf Stephan, May 26 2004
Row n of the array gives Taylor series expansion of F_n(t)/F_{n+1}(t), where F_n(t) are the Fibonacci polynomials defined in A259475 [Kreweras, 1970]. - N. J. A. Sloane, Jul 03 2015

			T(3,2) = 4 since the paths of length 2*3 (7 points) with all values less than or equal to 2 can take the routes 0101010, 0101210, 0121010 or 0121210, but not 0123210.
From _Peter Luschny_, Aug 27 2014: (Start)
Trees with n nodes and height <= h:
h\n  1  2  3  4   5   6    7    8     9    10     11
---------------------------------------------------------
[ 1] 1, 0, 0, 0,  0,  0,   0,   0,    0,    0,     0, ...  A063524
[ 2] 1, 1, 1, 1,  1,  1,   1,   1,    1,    1,     1, ...  A000012
[ 3] 1, 1, 2, 4,  8, 16,  32,  64,  128,  256,   512, ...  A011782
[ 4] 1, 1, 2, 5, 13, 34,  89, 233,  610, 1597,  4181, ...  A001519
[ 5] 1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281,  9842, ...  A124302
[ 6] 1, 1, 2, 5, 14, 42, 131, 417, 1341, 4334, 14041, ...  A080937
[ 7] 1, 1, 2, 5, 14, 42, 132, 428, 1416, 4744, 16016, ...  A024175
[ 8] 1, 1, 2, 5, 14, 42, 132, 429, 1429, 4846, 16645, ...  A080938
[ 9] 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861, 16778, ...  A033191
[10] 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, ...  A211216
---------------------------------------------------------
The generating functions are listed in A211216. Note that the values up to the main diagonal are the Catalan numbers A000108.
(End)

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Pattern-restricted cyclic permutations with a pattern-restricted cycle form, arXiv:2408.15000 [math.CO], 2024. See also Enum. Comb. Appl. (2025) Vol. 5, No. 1, Art. No. S2R3. See p. 7.
Nicolaas Govert de Bruijn, Donald E. Knuth, and Stephen O. Rice, The Average Height of Planted Plane Trees, Graph Theory and Computing, (1972) 15-22.
Yann Dijoux, Chebyshev polynomials involved in the Householder's method for square roots, arXiv:2501.04703 [math.GM], 2024. Mentions this sequence.
Andrzej Dudek, Jarosław Grytczuk, and Andrzej Ruciński, Largest bipartite sub-matchings of a random ordered matching or a problem with socks, arXiv:2402.02223 [math.CO], 2024.
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Aleksandar Ilic and Andreja Ilic, On the number of restricted Dyck paths, Filomat 25:3 (2011), 191-201; DOI: 10.2298/FIL1103191I.
Anthony Joseph and Polyxeni Lamprou, A new interpretation of Catalan numbers, arXiv preprint arXiv:1512.00406 [math.CO], 2015.
Myrto Kallipoliti, Robin Sulzgruber, and Eleni Tzanaki, Patterns in Shi tableaux and Dyck paths, arXiv:2006.06949 [math.CO], 2020.
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (page 10, Corollary 3).
Christian Krattenthaler, Permutations with restricted patterns and Dyck paths, arXiv:math/0002200 [math.CO], 2000.
Germain Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41. See page 38.
Germain Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
Erkko Lehtonen and Tamás Waldhauser, Associative spectra of graph algebras I. Foundations, undirected graphs, antiassociative graphs, arXiv:2011.07621 [math.CO], 2020. See also J. of Algebraic Combinatorics (2021) Vol. 53, 613-638.
Toufik Mansour, Restricted even permutations and Chebyshev polynomials, arXiv:math/0302014 [math.CO], 2003.
Toufik Mansour and Alek Vainshtein, Restricted permutations, continued fractions and Chebyshev polynomials, arXiv:math/9912052 [math.CO], 1999-2000.
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See pp. 5-6.
Thomas Tunstall, Tim Rogers, and Wolfram Möbius, Assisted percolation of slow-spreading mutants in heterogeneous environments, arXiv:2303.01579 [q-bio.PE], 2023.
Vince White, Enumeration of Lattice Paths with Restrictions, (2024). Electronic Theses and Dissertations. 2799. See pp. 20, 25.

Cf. A000108, A079214, A080935, A080936. Rows include A000012, A057427, A040000 (offset), columns include (essentially) A000007, A000012, A011782, A001519, A007051, A080937, A024175, A080938, A033191, A211216. Main diagonal is A000108.

Cf. A094718 (involutions). Cf. also A259475.

# As a triangular array:
b:= proc(x, y, k) option remember; `if`(y>min(k, x) or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, k)+ b(x-1, y+1, k)))
    end:
A:= (n, k)-> b(2*n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 06 2012
# As a square array:
A := proc(n,k) option remember; local j; if n = 1 then 1 elif k = 1 then 0 else add(A(n-j,k)*A(j,k-1), j=1..n-1) fi end:
linalg[matrix](10, 12, (n,k) -> A(k,n)); # Peter Luschny, Aug 27 2014

A[n_, k_] := A[n, k] = Which[n == 1, 1, k == 1, 0, True, Sum[A[n-j, k]*A[j, k-1], {j, 1, n-1}]]; Table[A[k-n+1, n], {k, 1, 13}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Peter Luschny *)

A(N, K) = {
  my(m = matrix(N, K, n, k, n==1));
  for (n = 2, N,
  for (k = 2, K,
       m[n,k] = sum(i = 1, n-1, m[n-i,k] * m[i,k-1])));
  return(m);
}
A(11,10)~  \\ Gheorghe Coserea, Jan 13 2016

T(n, k) = Sum_{0A080935(n, k) = T(n, k-1)+A080936(n, k); for k>=n T(n, k) = A000108(n).
T(n, k) = 2^(2n+1)/(k+2) * Sum_{i=1..k+1} (sin(Pi*i/(k+2))*cos(Pi*i/(k+2))^n)^2 for n>=1. - Herbert Kociemba, Apr 28 2004
G.f. of n-th row: B(n)/B(n+1) where B(j)=[(1+sqrt(1-4x))/2]^j-[(1-sqrt(1-4x))/2]^j.

A289481 Number A(n,k) of Dyck paths of semilength k*n and height n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 7, 1, 0, 1, 1, 31, 57, 1, 0, 1, 1, 127, 1341, 484, 1, 0, 1, 1, 511, 26609, 59917, 4199, 1, 0, 1, 1, 2047, 497845, 5828185, 2665884, 36938, 1, 0, 1, 1, 8191, 9096393, 517884748, 1244027317, 117939506, 328185, 1, 0

Offset: 0

Alois P. Heinz, Jul 06 2017

For fixed k > 1, A(n,k) ~ 2^(2*k*n + 3) * k^(2*k*n + 1/2) / ((k-1)^((k-1)*n + 1/2) * (k+1)^((k+1)*n + 7/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 14 2017

			Square array A(n,k) begins:
  1, 1,    1,       1,          1,            1, ...
  0, 1,    1,       1,          1,            1, ...
  0, 1,    7,      31,        127,          511, ...
  0, 1,   57,    1341,      26609,       497845, ...
  0, 1,  484,   59917,    5828185,    517884748, ...
  0, 1, 4199, 2665884, 1244027317, 517500496981, ...

Alois P. Heinz, Antidiagonals n = 0..80, flattened

Columns k=0..10 give: A000007, A000012, A268316, A289473, A289474, A289475, A289476, A289477, A289478, A289479, A289480.
Rows n=0-2 give: A000012, A057427, A083420(k+1).
Main diagonal gives A289482.
Cf. A080936.

b:= proc(x, y, k) option remember;
      `if`(x=0, 1, `if`(y>0, b(x-1, y-1, k), 0)+
      `if`(y <  min(x-1, k), b(x-1, y+1, k), 0))
    end:
A:= (n, k)-> `if`(n=0, 1, b(2*n*k, 0, n)-b(2*n*k, 0, n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_, k_]:=b[x, y, k]=If[x==0, 1, If[y>0, b[x - 1, y - 1, k], 0] + If[yIndranil Ghosh, Jul 07 2017, after Maple code *)

A291883 Number T(n,k) of symmetrically unique Dyck paths of semilength n and height k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 9, 11, 4, 1, 0, 1, 19, 31, 19, 5, 1, 0, 1, 35, 91, 69, 29, 6, 1, 0, 1, 71, 250, 252, 127, 41, 7, 1, 0, 1, 135, 690, 855, 540, 209, 55, 8, 1, 0, 1, 271, 1863, 2867, 2117, 1005, 319, 71, 9, 1, 0, 1, 527, 5017, 9339, 8063, 4411, 1705, 461, 89, 10, 1

Offset: 0

Alois P. Heinz, Sep 05 2017

			: T(4,2) = 5:       /\      /\        /\/\    /\  /\    /\/\/\
:              /\/\/  \  /\/  \/\  /\/    \  /  \/  \  /      \
:
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   2,   1;
  0, 1,   5,   3,   1;
  0, 1,   9,  11,   4,   1;
  0, 1,  19,  31,  19,   5,   1;
  0, 1,  35,  91,  69,  29,   6,  1;
  0, 1,  71, 250, 252, 127,  41,  7, 1;
  0, 1, 135, 690, 855, 540, 209, 55, 8, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A057427, A056326, A291887, A291888, A291889, A291890, A291891, A291892, A291893, A291894.
Main and first two lower diagonals give A000012, A001477, A028387(n-1) for n>0.
Row sums give A007123(n+1).
T(2n,n) give A291885.
Cf. A080936, A132890, A291886.

b:= proc(x, y, k) option remember; `if`(x=0, z^k, `if`(y0, b(x-1, y-1, k), 0))
    end:
g:= proc(x, y, k) option remember; `if`(x=0, z^k, `if`(y>0,
      g(x-2, y-1, k), 0)+ g(x-2, y+1, max(y+1, k)))
    end:
T:= n-> (p-> seq(coeff(p, z, i)/2, i=0..n))(b(2*n, 0$2)+g(2*n, 0$2)):
seq(T(n), n=0..14);

b[x_, y_, k_] := b[x, y, k] = If[x == 0, z^k, If[y < x - 1, b[x - 1, y + 1, Max[y + 1, k]], 0] + If[y > 0, b[x - 1, y - 1, k], 0]];
g[x_, y_, k_] := g[x, y, k] = If[x == 0, z^k, If[y > 0, g[x - 2, y - 1, k], 0] + g[x - 2, y + 1, Max[y + 1, k]]];
T[n_] := Function[p, Table[Coefficient[p, z, i]/2, {i, 0, n}]][b[2*n, 0, 0] + g[2*n, 0, 0]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

from sympy.core.cache import cacheit
from sympy import Poly, Symbol, flatten
z=Symbol('z')
@cacheit
def b(x, y, k): return z**k if x==0 else (b(x - 1, y + 1, max(y + 1, k)) if y0 else 0)
@cacheit
def g(x, y, k): return z**k if x==0 else (g(x - 2, y - 1, k) if y>0 else 0) + g(x - 2, y + 1, max(y + 1, k))
def T(n): return 1 if n==0 else [i//2 for i in Poly(b(2*n, 0, 0) + g(2*n, 0, 0)).all_coeffs()[::-1]]
print(flatten(map(T, range(15)))) # Indranil Ghosh, Sep 06 2017

T(n,k) = (A080936(n,k) + A132890(n,k))/2.

Sum_{k=1..n} k * T(n,k) = A291886(n).

A358729 Difference between the number of nodes and the node-height of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 1, 3, 2, 2, 3, 1, 2, 3, 3, 2, 4, 3, 1, 3, 0, 4, 2, 2, 3, 4, 2, 3, 3, 3, 1, 4, 2, 2, 4, 3, 2, 4, 4, 4, 3, 3, 3, 5, 3, 4, 4, 2, 1, 4, 3, 1, 5, 5, 4, 3, 2, 3, 4, 4, 2, 5, 3, 3, 5, 4, 3, 4, 1, 4, 6, 2, 2, 5, 4, 3, 3, 3, 3, 5, 4, 4, 2, 3, 4, 5, 3, 5, 4, 5, 2, 4, 4, 4, 5, 4, 3, 6

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Node-height is the number of nodes in the longest path from root to leaf.
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.
Because the number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n, i.e., A317713(n) (= 1+A324923(n)), is always at least one larger than the depth of the same tree (= A109082(n)), it follows that a(n) >= A366386(n) for all n. - Antti Karttunen, Oct 23 2023

			The tree (oo(oo(o))) with Matula-Goebel number 148 has 8 nodes and node-height 4, so a(148) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to Matula-Goebel numbers

Positions of 0's are A007097.
Positions of first appearances are A358730.
Positions of 1's are A358731.
Other differences: A358580, A358724, A358726.
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.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A056239, A112798.
Cf. A206487, A209638, A316321, A358576, A358577, A358592, A358725, A366386.

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

A061775(n) = if(1==n, 1, if(isprime(n), 1+A061775(primepi(n)), {my(pfs,t,i); pfs=factor(n); pfs[,1]=apply(t->A061775(t),pfs[,1]); (1-bigomega(n)) + sum(i=1, omega(n), pfs[i,1]*pfs[i,2])}));
A358552(n) = { my(v=factor(n)[, 1], d=0); while(#v, d++; v=fold(setunion, apply(p->factor(primepi(p))[, 1]~, v))); (1+d); }; \\ (after program given in A109082 by Kevin Ryde, Sep 21 2020)
A358729(n) = (A061775(n)-A358552(n)); \\ Antti Karttunen, Oct 23 2023

a(n) = A061775(n) - A358552(n).

a(n) = A196050(n) - A109082(n). - Antti Karttunen, Oct 23 2023

Data section extended up to a(108) by Antti Karttunen, Oct 23 2023

A358591 Number of 2n-node rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

Original entry on oeis.org

0, 0, 2, 17, 94, 464, 2162, 9743, 42962, 186584, 801316, 3412034, 14430740, 60700548, 254180426, 1060361147, 4409342954, 18285098288, 75645143516, 312286595342, 1286827096964, 5293833371408, 21745951533236, 89208948855542, 365523293690804, 1496048600896784

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(3) = 2 and a(4) = 17 trees:
  ((o)(oo))  (((o))(ooo))
  (o(o)(o))  (((o)(ooo)))
             (((oo))(oo))
             (((oo)(oo)))
             ((o)((ooo)))
             ((o)(o(oo)))
             ((o)(oo(o)))
             ((o(o)(oo)))
             ((oo)(o(o)))
             ((oo(o)(o)))
             (o((o))(oo))
             (o((o)(oo)))
             (o(o)((oo)))
             (o(o)(o(o)))
             (o(o(o)(o)))
             (oo((o)(o)))
             (oo(o)((o)))

Andrew Howroyd, Table of n, a(n) for n = 1..100

For leaves = internals we have A185650 aerated, ranked by A358578.
For height = internals we have A358587, ranked by A358576, ordered A358588.
For height = leaves we have A358589, ranked by A358577, ordered A358590.
These trees are ranked by A358592.
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.
Cf. A065097, A109082, A109129, A342507, A358552, A358581-A358586.

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}]==Depth[#]-1&]],{n,2,15,2}]

\\ Needs R(n,f) defined in A358589.
seq(n) = {Vecrev(R(2*n, (h,p)->if(h<=n, x^h*polcoef(polcoef(p, 2*h, x), h, y))), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023

A358585 Number of ordered rooted trees with n nodes, most of which are leaves.

Original entry on oeis.org

1, 0, 1, 1, 7, 11, 66, 127, 715, 1549, 8398, 19691, 104006, 258194, 1337220, 3467115, 17678835, 47440745, 238819350, 659060677, 3282060210, 9271024542, 45741281820, 131788178171, 644952073662, 1890110798926, 9183676536076, 27316119923002, 131873975875180, 397407983278484

Offset: 1

Gus Wiseman, Nov 24 2022

nonn

			The a(1) = 1 through a(6) = 11 ordered trees:
  o  .  (oo)  (ooo)  (oooo)   (ooooo)
                     ((o)oo)  ((o)ooo)
                     ((oo)o)  ((oo)oo)
                     ((ooo))  ((ooo)o)
                     (o(o)o)  ((oooo))
                     (o(oo))  (o(o)oo)
                     (oo(o))  (o(oo)o)
                              (o(ooo))
                              (oo(o)o)
                              (oo(oo))
                              (ooo(o))

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

For equality we have A000891, unordered A185650.
Odd-indexed terms are A065097.
The unordered version is A358581.
The opposite is the same, unordered A358582.
The non-strict version is A358586, unordered A358583.
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, A358584, A358588, A358590.

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,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-1)/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

A258109 Number of balanced parenthesis expressions of length 2n and depth 3.

Original entry on oeis.org

1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673, 62721698, 164531565, 431397285, 1130708866, 2962826465, 7761964833, 20331456642, 53249182309, 139449644717, 365166860706, 956185155129, 2503657040137

Offset: 3

Gheorghe Coserea, May 20 2015

a(n) is the number of Dyck paths of length 2n and height 3. For example, a(3) = 1 because there is only one such Dyck path which is UUUDDD. - Ran Pan, Sep 26 2015

a(n) is the number of rooted plane trees with n+1 nodes and height 3 (see example for correspondence). - Gheorghe Coserea, Feb 04 2016

			For n=4, the a(4) = 5 solutions are
                /\       /\
               /  \        \
LRLLLRRR    /\/    \        \
................................
                /\        |
             /\/  \      / \
LLRLLRRR    /      \        \
................................
              /\/\        |
             /    \       |
LLLRLRRR    /      \     / \
................................
              /\          |
             /  \/\      / \
LLLRRLRR    /      \    /
................................
              /\          /\
             /  \        /
LLLRRRLR    /    \/\    /

S. S. Skiena and M. A. Revilla, Programming Challenges: The Programming Contest Training Manual, Springer, 2006, page 140.

Alois P. Heinz, Table of n, a(n) for n = 3..1000 (first 40 terms from Gheorghe Coserea)
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (5,-7,2).

Column k=3 of A080936.
Column k=2 of A287213.
Cf. A000045, A000079, A262600.

typedef long long unsigned Integer;
Integer a(int n)
{
    int i;
    Integer pow2 = 1, a[3] = {0};
    for (i = 3; i <= n; ++i) {
        a[ i%3 ] = pow2 + 3 * a[ (i-1)%3 ] - a[ (i-2)%3 ];
        pow2 = pow2 * 2;
    }
    return a[ (i-1)%3 ];
}

I:=[1,5,18,57,169]; [n le 5 select I[n] else 5*Self(n-1) - 7*Self(n-2) + 2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Sep 26 2015

a:= proc(n) option remember; `if`(n<3, 0,
      `if`(n=3, 1, 2^(n-3) +3*a(n-1) -a(n-2)))
    end:
seq(a(n), n=3..30);  # Alois P. Heinz, May 20 2015

Join[{1, 5}, LinearRecurrence[{5, -7, 2}, {18, 57, 169}, 30]] (* Vincenzo Librandi, Sep 26 2015 *)

Vec(-x^3/((2*x-1)*(x^2-3*x+1)) + O(x^100)) \\ Colin Barker, May 24 2015

a(n) = fibonacci(2*n-1) - 2^(n-1)  \\ Gheorghe Coserea, Feb 04 2016

a(n) = 2^(n-3) + 3 * a(n-1) - a(n-2).
a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3) for n>5. - Colin Barker, May 24 2015
G.f.: -x^3 / ((2*x-1)*(x^2-3*x+1)). - Colin Barker, May 24 2015
a(n) = A000045(2n-1) - A000079(n-1). - Gheorghe Coserea, Feb 04 2016
a(n) = 2^(-1-n)*(-5*4^n - (-5+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(5+sqrt(5))) / 5. - Colin Barker, Jun 05 2017
a(n) = Sum_{i=1..n-1} A061667(i)*(n-1-i) - Tim C. Flowers, May 16 2018

A358724 Difference between the number of internal (non-leaf) nodes and the edge-height of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 0, 1, 2, 1, 0, 0, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 0, 2, 1, 0, 0, 2, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 3, 0, 1, 1, 2, 0, 1

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Edge-height (A109082) is the number of edges in the longest path from root to leaf.

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 tree (o(o)((o))(oo)) with Matula-Goebel number 210 has edge-height 3 and 5 internal nodes, so a(210) = 2.

Positions of 0's are A209638, complement A358725.
Positions of 1's are A358576, counted by A358587.
Other differences: A358580, A358726, A358729.
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.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A185650, A206487, A316321, A358577, A358578, A358592, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],[_],{0,Infinity}]-(Depth[MGTree[n]]-2),{n,100}]

a(n) = A342507(n) - A109082(n).

A358726 Difference between the node-height and the number of leaves in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 29 2022

sign

Node-height is the number of nodes in the longest path from root to leaf.

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 tree (oo(oo(o))) with Matula-Goebel number 148 has node-height 4 and 5 leaves, so a(148) = -1.

Positions of first appearances are A007097 and latter terms of A000079.
Positions of 0's are A358577.
Other differences: A358580, A358724, A358729.
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.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A056239, A112798.
Cf. A185650, A206487, A209638, A358576, A358578, A358587, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[(Depth[MGTree[n]]-1)-Count[MGTree[n],{},{0,Infinity}],{n,1000}]

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

cp's OEIS Frontend

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

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A080934 Square array read by antidiagonals of number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2n steps with all values less than or equal to k.

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A289481 Number A(n,k) of Dyck paths of semilength k*n and height n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A291883 Number T(n,k) of symmetrically unique Dyck paths of semilength n and height k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A358729 Difference between the number of nodes and the node-height of the rooted tree with Matula-Goebel number n.

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A358591 Number of 2n-node rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

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A358585 Number of ordered rooted trees with n nodes, most of which are leaves.

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