A001190 -id:A001190 - cp's OEIS frontend

A292050 Matula-Goebel numbers of semi-binary rooted trees.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 39, 41, 43, 46, 47, 49, 51, 55, 58, 59, 62, 65, 69, 73, 77, 79, 82, 83, 85, 86, 87, 91, 93, 94, 97, 101, 109, 115, 118, 119, 121, 123, 127, 129, 137, 139, 141, 143, 145

Offset: 1

Gus Wiseman, Sep 08 2017

nonn

An unlabeled rooted tree is semi-binary if all out-degrees are <= 2. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

Cf. A000081, A001190, A007097, A036656, A061773, A111299, A214577, A291636.

nn=200;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
semibinQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]<=2,And@@semibinQ/@m]]];
Select[Range[nn],semibinQ]

A298426 Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 11, 4, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 23, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 19 2018

Row sums are A298422.

			Triangle begins:
1
0  1
0  1  1
0  1  0  1
0  1  1  0  1
0  1  0  0  0  1
0  1  2  1  0  0  1
0  1  0  0  0  0  0  1
0  1  3  0  1  0  0  0  1
0  1  0  2  0  0  0  0  0  1
0  1  6  0  0  1  0  0  0  0  1
0  1  0  0  0  0  0  0  0  0  0  1
0  1  11 4  2  0  1  0  0  0  0  0  1
0  1  0  0  0  0  0  0  0  0  0  0  0  1
0  1  23 0  0  0  0  1  0  0  0  0  0  0  1
0  1  0  8  0  2  0  0  0  0  0  0  0  0  0  1

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A000005, A000081, A000598, A001190, A001678, A003238, A004111, A067538, A143773, A289078, A289079, A295461, A298118, A298422, A298423, A298424.

nn=16;
arut[n_,k_]:=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[arut[#,k]&/@c]]]/@Select[IntegerPartitions[n-1],Length[#]===k&]]
Table[arut[n,k]//Length,{n,nn},{k,0,n-1}]

A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

Original entry on oeis.org

1, 4, 14, 14, 86, 86, 49, 86, 86, 886, 886, 454, 886, 886, 301, 301, 301, 886, 886, 301, 454, 886, 886, 13766, 13766, 6418, 13766, 13766, 3986, 3986, 3986, 13766, 13766, 3986, 6418, 13766, 13766, 3101, 3101, 1589, 3101, 3101, 1849, 1849, 3101, 13766

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted binary trees encoded in range [A014137(n-1)..A014138(n-1)] of A014486 to A001190(n+1) non-oriented rooted binary trees, encoded by their Matula-Goebel numbers (when viewed as a subset of non-oriented rooted general trees). See also the comments at A127301.
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127302(SP(n)) = A127302(n) for all n, then it preserves the non-oriented form of a binary tree. Examples of such automorphisms include A069770, A057163, A122351, A069767/A069768, A073286-A073289, A089854, A089859/A089863, A089864, A122282, A123492-A123494, A123715/A123716, A127377-A127380, A127387 and A127388.
A153835 divides natural numbers to same equivalence classes, i.e. a(i) = a(j) <=> A153835(i) = A153835(j) - Antti Karttunen, Jan 03 2013

			A001190(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, terms A014486(4..8) encode the following five plane binary trees:
........\/.....\/.................\/.....\/...
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
n=.....4........5........6........7........8..
The trees in positions 4, 5, 7 and 8 all produce Matula-Goebel number A000040(1)*A000040(A000040(1)*A000040(A000040(1)*A000040(1))) = 2*A000040(2*A000040(2*2)) = 2*A000040(14) = 2*43 = 86, as they are just different planar representations of the one and same non-oriented tree. The tree in position 6 produces Matula-Goebel number A000040(A000040(1)*A000040(1)) * A000040(A000040(1)*A000040(1)) = A000040(2*2) * A000040(2*2) = 7*7 = 49. Thus a(4..8) = 86,86,49,86,86.

Index entries for sequences related to Matula-Goebel numbers

Cf A127301, A153835, A153829.

a(n) = A127301(A057123(n)).

Can be also computed directly as a fold, see the Scheme-program. - Antti Karttunen, Jan 03 2013

A317707 Number of powerful rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 13, 22, 29, 46, 57, 94, 115, 180, 230, 349, 435, 671, 830, 1245, 1572, 2320, 2894, 4287, 5328, 7773, 9752, 14066, 17547, 25328, 31515, 45010, 56289, 79805, 99467, 140778, 175215, 246278, 307273, 429421, 534774, 745776, 927776, 1287038

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

An unlabeled rooted tree is powerful if either it is a single node or a single node with a single powerful tree as a branch, or if the branches of the root all appear with multiplicities greater than 1 and are themselves powerful trees.

			The a(7) = 11 powerful rooted trees:
  ((((((o))))))
  (((((oo)))))
  ((((ooo))))
  ((((o)(o))))
  (((oooo)))
  ((ooooo))
  (((o))((o)))
  ((oo)(oo))
  ((o)(o)(o))
  (oo(o)(o))
  (oooooo)

Alois P. Heinz, Table of n, a(n) for n = 1..8000

Cf. A000081, A001190, A001694, A004111, A301700, A303431, A317102.

Cf. A317705, A317708, A317709, A317710, A317711, A317712, A317718, A317719.

h:= proc(n, k, t) option remember; `if`(k=0, binomial(n+t, t),
      `if`(n=0, 0, add(h(n-1, k-j, t+1), j=2..k)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))
    end:
a:= proc(n) option remember; `if`(n<2, n, b(n-1$2)+a(n-1)) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 31 2018

purt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Or[Length[#]==1,Min@@Length/@Split[#]>1]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,10}]
(* Second program: *)
h[n_, k_, t_] := h[n, k, t] = If[k == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, k - j, t + 1], {j, 2, k}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]* h[a[i], j, 0], {j, 0, n/i}]]];
a[n_] := a[n] = If[n < 2, n, b[n - 1, n - 1] + a[n - 1]];
Array[a, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(27)-a(45) from Alois P. Heinz, Aug 31 2018

A032027 Number of planted planar trees (n+1 nodes) where any 2 subtrees extending from the same node are different.

Original entry on oeis.org

1, 1, 1, 3, 5, 13, 35, 95, 255, 715, 2081, 6003, 17645, 52127, 155863, 468129, 1415521, 4301055, 13134789, 40275109, 123970669, 382919917, 1186475687, 3686899725, 11487023793, 35876838669, 112304155021, 352276801491

Offset: 1

Christian G. Bower

			From _Gus Wiseman_, Nov 15 2022: (Start)
The a(1) = 1 through a(6) = 13 ordered rooted identity trees (ranked by A358374):
  o  (o)  ((o))  ((o)o)   (((o))o)   (((o)o)o)
                 (o(o))   (((o)o))   ((o(o))o)
                 (((o)))  ((o(o)))   (o((o)o))
                          (o((o)))   (o(o(o)))
                          ((((o))))  ((((o)))o)
                                     ((((o))o))
                                     ((((o)o)))
                                     (((o))(o))
                                     (((o(o))))
                                     ((o)((o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to rooted trees

The unordered version is A004111, ranked by A276625.
These trees (ordered rooted identity) are ranked by A358374.
Cf. A000081, A001190, A005043, A003238, A063895, A358377.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,[_]?(!UnsameQ@@#&)]&]],{n,1,10}] (* Gus Wiseman, Nov 15 2022 *)

AGK(v)={apply(p->subst(serlaplace(y^0*p),y,1), Vec(prod(k=1, #v, (1 + x^k*y + O(x*x^#v))^v[k])-1, -#v))}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], AGK(v))); v} \\ Andrew Howroyd, Sep 20 2018

Shifts left under "AGK" (ordered, elements, unlabeled) transform.

A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 4, 6, 1, 0, 1, 1, 1, 2, 4, 8, 11, 1, 0, 1, 1, 1, 2, 4, 9, 17, 23, 1, 0, 1, 1, 1, 2, 4, 9, 19, 39, 46, 1, 0, 1, 1, 1, 2, 4, 9, 20, 45, 89, 98, 1, 0, 1, 1, 1, 2, 4, 9, 20, 47, 106, 211, 207, 1, 0

Offset: 0

Alois P. Heinz, Feb 01 2018

			Square array A(n,k) begins:
  1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   2,   2,   2,   2,   2,   2,   2,   2,   2, ...
  0, 1,   3,   4,   4,   4,   4,   4,   4,   4,   4, ...
  0, 1,   6,   8,   9,   9,   9,   9,   9,   9,   9, ...
  0, 1,  11,  17,  19,  20,  20,  20,  20,  20,  20, ...
  0, 1,  23,  39,  45,  47,  48,  48,  48,  48,  48, ...
  0, 1,  46,  89, 106, 112, 114, 115, 115, 115, 115, ...
  0, 1,  98, 211, 260, 277, 283, 285, 286, 286, 286, ...
  0, 1, 207, 507, 643, 693, 710, 716, 718, 719, 719, ...

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

Columns k=1-11 give: A000012, A001190(n+1), A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556.
Main diagonal gives A000081 for n>0.
A(2n,n) gives A299039.
Cf. A244372.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
A:= (n, k)-> `if`(n=0, 1, b(n-1$2, k$2)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[ b[i-1, i-1, k, k]+j-1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]];
A[n_, k_] := If[n == 0, 1, b[n - 1, n - 1, k, k]];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)

from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, i, t, k): return 1 if n==0 else 0 if i<1 else sum([binomial(b(i-1, i-1, k, k)+j-1, j)*b(n-i*j, i-1, t-j, k) for j in range(min(t, n//i)+1)])
def A(n, k): return 1 if n==0 else b(n-1, n-1, k, k)
for d in range(15): print([A(n, d-n) for n in range(d+1)]) # Indranil Ghosh, Mar 02 2018, after Maple code

A(n,k) = Sum_{i=0..k} A244372(n,i) for n>0, A(0,k) = 1.

A317708 Number of aperiodic relatively prime trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 20, 48, 108, 255, 595, 1435, 3434, 8372, 20419, 50289, 124289, 309122, 771508, 1934462

Offset: 1

Gus Wiseman, Aug 05 2018

An unlabeled rooted tree is aperiodic and relatively prime iff either it is a single node or a single node with a single aperiodic relatively prime branch, or the branches directly under any given node have empty intersection (relatively prime) and also have relatively prime multiplicities (aperiodic) and are themselves aperiodic relatively prime trees.

			The a(6) = 10 aperiodic relatively prime trees:
  (((((o)))))
  (((o(o))))
  ((o((o))))
  ((oo(o)))
  (o(((o))))
  (o(o(o)))
  ((o)((o)))
  (oo((o)))
  (o(o)(o))
  (ooo(o))

Gus Wiseman, All 48 aperiodic relatively prime trees with 8 nodes.

Cf. A000081, A001190, A004111, A301700, A303431, A316501, A316500.

Cf. A317705, A317707, A317709, A317710, A317711, A317712.

rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,And[Intersection@@#=={},GCD@@Length/@Split[#]==1]]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

A339888 Number of non-isomorphic multiset partitions of weight n into singletons or strict pairs.

Original entry on oeis.org

1, 1, 3, 5, 13, 23, 55, 104, 236, 470, 1039, 2140, 4712, 9962, 21961, 47484, 105464, 232324, 521338, 1167825, 2651453, 6031136, 13863054, 31987058, 74448415, 174109134, 410265423, 971839195, 2317827540, 5558092098, 13412360692, 32542049038, 79424450486

Offset: 0

Gus Wiseman, Jan 09 2021

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 13 multiset partitions:
  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{1,2}}
         {{1},{1}}  {{2},{1,2}}    {{1,2},{3,4}}
         {{1},{2}}  {{1},{1},{1}}  {{1,3},{2,3}}
                    {{1},{2},{2}}  {{1},{1},{2,3}}
                    {{1},{2},{3}}  {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The version for set partitions is A000085, with ordered version A080599.
The case of integer partitions is 1 + A004526(n), ranked by A003586.
Non-isomorphic multiset partitions are counted by A007716.
The case without singletons is A007717.
The version allowing non-strict pairs (x,x) is A320663.
A001190 counts rooted trees with out-degrees <= 2, ranked by A292050.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
A339887 counts factorizations into primes or squarefree semiprimes.
Cf. A001055, A007718, A316983, A319616, A320656, A321729, A339740, A339741.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
gs(v) = {sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i], v[j])); g*x^(2*v[i]*v[j]/g))) + sum(i=1, #v, my(r=v[i]); (1 + (1+r)%2)*x^r + ((r-1)\2)*x^(2*r))}
a(n)={if(n==0, 1, my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(gs(p) + O(x*x^n), -n))[n]); s/n!)} \\ Andrew Howroyd, Apr 16 2021

Terms a(11) and beyond from Andrew Howroyd, Apr 16 2021

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}]

A292085 Number A(n,k) of (unlabeled) rooted trees with n leaf nodes and without unary nodes or outdegrees larger than k; 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, 2, 2, 0, 1, 1, 2, 4, 3, 0, 1, 1, 2, 5, 9, 6, 0, 1, 1, 2, 5, 11, 23, 11, 0, 1, 1, 2, 5, 12, 30, 58, 23, 0, 1, 1, 2, 5, 12, 32, 80, 156, 46, 0, 1, 1, 2, 5, 12, 33, 87, 228, 426, 98, 0, 1, 1, 2, 5, 12, 33, 89, 251, 656, 1194, 207, 0

Offset: 1

Alois P. Heinz, Sep 08 2017

			:               T(4,3) = 4             :
:                                      :
:       o       o         o       o    :
:      / \     / \       / \     /|\   :
:     o   N   o   o     o   N   o N N  :
:    / \     ( ) ( )   /|\     ( )     :
:   o   N    N N N N  N N N    N N     :
:  ( )                                 :
:  N N                                 :
:                                      :
Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,   2,   2,   2,   2,   2,   2, ...
  0,  2,   4,   5,   5,   5,   5,   5, ...
  0,  3,   9,  11,  12,  12,  12,  12, ...
  0,  6,  23,  30,  32,  33,  33,  33, ...
  0, 11,  58,  80,  87,  89,  90,  90, ...
  0, 23, 156, 228, 251, 258, 260, 261, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to rooted trees

Columns k=1-10 give: A063524, A001190, A268172, A292210, A292211, A292212, A292213, A292214, A292215, A292216.
Main diagonal gives A000669.
Cf. A244372, A288942, A292086.

b:= proc(n, i, v, k) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n

b[n_, i_, v_, k_] := b[n, i, v, k] = If[n == 0, If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0, If[v == n, 1, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, v - j, k], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n < 2, n, Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];
Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

A(n,k) = Sum_{j=1..k} A292086(n,j).

cp's OEIS Frontend

A292050 Matula-Goebel numbers of semi-binary rooted trees.

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A298426 Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.

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Mathematica

A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

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A317707 Number of powerful rooted trees with n nodes.

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Maple

Mathematica

Extensions

A032027 Number of planted planar trees (n+1 nodes) where any 2 subtrees extending from the same node are different.

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Mathematica

PARI

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A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

Mathematica

Python

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A317708 Number of aperiodic relatively prime trees with n nodes.

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Mathematica

A339888 Number of non-isomorphic multiset partitions of weight n into singletons or strict pairs.

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