A055277 -id:A055277 - cp's OEIS frontend

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

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

A358379 Edge-height (or depth) of the n-th standard ordered rooted tree.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 2, 2, 3, 2, 2, 1, 4, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 1, 3, 4, 2, 2, 3, 3, 3, 3, 2, 3, 2, 2, 3, 2, 2, 2, 4, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 1, 3, 3, 4, 4, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 2, 3, 3, 3, 2, 2

Offset: 1

Gus Wiseman, Nov 16 2022

nonn

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The standard ordered rooted tree ranking begins:
  1: o        10: (((o))o)   19: (((o))(o))
  2: (o)      11: ((o)(o))   20: (((o))oo)
  3: ((o))    12: ((o)oo)    21: ((o)((o)))
  4: (oo)     13: (o((o)))   22: ((o)(o)o)
  5: (((o)))  14: (o(o)o)    23: ((o)o(o))
  6: ((o)o)   15: (oo(o))    24: ((o)ooo)
  7: (o(o))   16: (oooo)     25: (o(oo))
  8: (ooo)    17: ((((o))))  26: (o((o))o)
  9: ((oo))   18: ((oo)o)    27: (o(o)(o))
For example, the 52nd ordered tree is (o((o))oo) so a(52) = 3.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Records occur at A004249.
The triangle counting trees by this statistic is A080936, unordered A034781.
Unordered version is A109082, nodes A061775, leaves A109129, edges A196050.
Leaves are counted by A358371.
Nodes are counted by A358372.
Node-height is a(n) + 1, unordered version is A358552.
A000081 counts unordered rooted trees, ranked by A358378.
A000108 counts ordered rooted trees.
A001263 counts ordered rooted trees by leaves.
Cf. A005043, A055277, A187306, A358373, A358374, A358375, A358376, A358377.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Table[Depth[srt[n]]-2,{n,100}]

A358575 Triangle read by rows where T(n,k) is the number of unlabeled n-node rooted trees with k = 0..n-1 internal (non-leaf) nodes.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 6, 1, 0, 1, 5, 14, 18, 9, 1, 0, 1, 6, 21, 39, 35, 12, 1, 0, 1, 7, 30, 72, 97, 62, 16, 1, 0, 1, 8, 40, 120, 214, 212, 103, 20, 1, 0, 1, 9, 52, 185, 416, 563, 429, 161, 25, 1

Offset: 1

Gus Wiseman, Nov 23 2022

			Triangle begins:
    1
    0    1
    0    1    1
    0    1    2    1
    0    1    3    4    1
    0    1    4    8    6    1
    0    1    5   14   18    9    1
    0    1    6   21   39   35   12    1
    0    1    7   30   72   97   62   16    1
    0    1    8   40  120  214  212  103   20    1
    0    1    9   52  185  416  563  429  161   25    1

Row sums are A000081.
Column k = n - 2 appears to be A002620.
Column k = 3 appears to be A006578.
The version for height instead of internal nodes is A034781.
Equals A055277 with rows reversed.
The ordered version is A090181 or A001263.
The central column is A185650, ordered A000891.
The left half sums to A358583, strict A358581.
The right half sums to A358584, strict A358582.
Cf. A000108, A065097, A109129, A342507, A358578, A358587, A358589.

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}]==k&]],{n,1,10},{k,0,n-1}]

A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.

Original entry on oeis.org

2, 6, 7, 9, 20, 22, 23, 26, 27, 29, 35, 41, 66, 76, 78, 79, 84, 86, 87, 90, 91, 93, 97, 102, 103, 106, 107, 109, 115, 117, 130, 136, 138, 139, 141, 146, 153, 163, 169, 193, 196, 197, 201, 226, 241, 262, 263, 296, 300, 302, 303, 308, 310, 311, 314, 315, 317

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The terms together with their corresponding rooted trees begin:
   2: (o)
   6: (o(o))
   7: ((oo))
   9: ((o)(o))
  20: (oo((o)))
  22: (o(((o))))
  23: (((o)(o)))
  26: (o(o(o)))
  27: ((o)(o)(o))
  29: ((o((o))))
  35: (((o))(oo))
  41: (((o(o))))
  66: (o(o)(((o))))
  76: (oo(ooo))
  78: (o(o)(o(o)))
  79: ((o(((o)))))
  84: (oo(o)(oo))
  86: (o(o(oo)))

These ordered trees are counted by A000891.
The unordered version is A358578, counted by A185650.
Height instead of leaves: counted by A358588, unordered A358576.
Height instead of internals: counted by A358590, unordered A358577.
Standard ordered tree number statistics: A358371, A358372, A358379, A358553.
A000081 counts rooted trees, ordered A000108.
A055277 counts trees by nodes and leaves, ordered A001263.
Cf. A014221, A206487, A358373, A358580, A358587, A358589.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],Count[srt[#],{},{0,Infinity}]==Count[srt[#],[_],{0,Infinity}]&]

A358371(a(n)) = A358553(a(n)).

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 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:
   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)).

A301342 Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 4, 1, 0, 0, 0, 1, 6, 5, 0, 0, 0, 0, 1, 9, 13, 2, 0, 0, 0, 0, 1, 12, 28, 11, 0, 0, 0, 0, 0, 1, 16, 53, 40, 3, 0, 0, 0, 0, 0, 1, 20, 91, 109, 26, 0, 0, 0, 0, 0, 0, 1, 25, 146, 254, 116, 6, 0, 0, 0, 0, 0, 0, 1, 30, 223, 524, 387, 61, 0, 0, 0, 0, 0, 0, 0, 1, 36

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1   0
1   0   0
1   1   0   0
1   2   0   0   0
1   4   1   0   0   0
1   6   5   0   0   0   0
1   9  13   2   0   0   0   0
1  12  28  11   0   0   0   0   0
1  16  53  40   3   0   0   0   0   0
1  20  91 109  26   0   0   0   0   0   0
1  25 146 254 116   6   0   0   0   0   0   0
1  30 223 524 387  61   0   0   0   0   0   0   0
The T(6,2) = 4 rooted identity trees: (((o(o)))), ((o((o)))), (o(((o)))), ((o)((o))).

A version with the zeroes removed is A055327.

Cf. A000081, A001190, A003238, A004111, A032305, A055277, A273873, A276625, A277098, A290689, A298118, A298422, A298426, A301343, A301344, A301345.

irut[n_]:=irut[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[irut/@c]],UnsameQ@@#&]]/@IntegerPartitions[n-1]];
Table[Length[Select[irut[n],Count[#,{},{-2}]===k&]],{n,8},{k,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

A301345 Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 2, 4, 1, 0, 0, 0, 0, 3, 4, 5, 1, 0, 0, 0, 0, 2, 6, 6, 6, 1, 0, 0, 0, 0, 1, 6, 10, 9, 7, 1, 0, 0, 0, 0, 1, 5, 12, 16, 12, 8, 1, 0, 0, 0, 0, 0, 4, 13, 22, 23, 16, 9, 1, 0, 0, 0, 0, 0, 3, 14, 27, 36, 32, 20, 10, 1, 0, 0, 0, 0, 0, 2, 11

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1   0
0   1   0
0   1   1   0
0   0   2   1   0
0   0   1   3   1   0
0   0   1   2   4   1   0
0   0   0   3   4   5   1   0
0   0   0   2   6   6   6   1   0
0   0   0   1   6  10   9   7   1   0
0   0   0   1   5  12  16  12   8   1   0
The T(9,5) = 6 transitive rooted trees: (o(o)(oo(o))), (o((oo))(oo)), (oo(o)(o(o))), (o(o)(o)(oo)), (ooo(o)((o))), (oo(o)(o)(o)).

Row sums are A290689.

Cf. A000081, A001190, A003238, A004111, A032305, A055277, A276625, A279861, A290760, A290822, A298422, A298426, A301342, A301343, A301344.

rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];
trat[n_]:=Select[rut[n],Complement[Union@@#,#]==={}&];
Table[Length[Select[trat[n],Count[#,{},{-2}]===k&]],{n,15},{k,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

A301364 Regular triangle where T(n,k) is the number of enriched p-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 4, 5, 1, 2, 6, 11, 12, 1, 3, 10, 26, 38, 34, 1, 3, 13, 39, 87, 117, 92, 1, 4, 19, 69, 181, 339, 406, 277, 1, 4, 23, 95, 303, 707, 1198, 1311, 806, 1, 5, 30, 143, 514, 1430, 2970, 4525, 4522, 2500, 1, 5, 35, 184, 762, 2446, 6124, 11627

Offset: 1

Gus Wiseman, Mar 19 2018

An enriched p-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

			Triangle begins:
  1
  1   1
  1   1   2
  1   2   4   5
  1   2   6  11  12
  1   3  10  26  38  34
  1   3  13  39  87 117  92
  1   4  19  69 181 339 406 277
  ...
The T(5,4) = 11 enriched p-trees: (((21)1)1), ((2(11))1), (((11)2)1), ((211)1), ((21)(11)), (((11)1)2), ((111)2), ((21)11), (2(11)1), ((11)21), (2111).

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

Last entries of each row give A196545. Row sums are A289501.

Cf. A008284, A055277, A063834, A220418, A273866, A273873, A281145, A290261, A299201, A299202, A299203, A300354, A300442, A300443, A301365-A301368.

eptrees[n_]:=Prepend[Join@@Table[Tuples[eptrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[Select[eptrees[n],Count[#,_Integer,{-1}]===k&]],{n,8},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); apply(p->Vecrev(p/y), v)}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

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

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A358379 Edge-height (or depth) of the n-th standard ordered rooted tree.

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A358575 Triangle read by rows where T(n,k) is the number of unlabeled n-node rooted trees with k = 0..n-1 internal (non-leaf) nodes.

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A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.

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Mathematica

Formula

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

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Mathematica

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A301342 Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.

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Mathematica

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

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Mathematica

PARI

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A301345 Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.

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Mathematica