A000081 -id:A000081 - cp's OEIS frontend

A342507 Number of internal nodes in rooted tree with Matula-Goebel number n.

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

Offset: 1

François Marques, Mar 14 2021

nonn

The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product_{T_i} prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration.)

			a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = 1 because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.

François Marques, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Matula-Goebel numbers

Other statistics are: A061775 (nodes), A109082 (edge-height), A109129 (leaves), A196050 (edges), A358552 (node-height).
An ordered version is A358553.
Positions of first appearances are A358554.
A000081 counts rooted trees, ordered A000108.
A358575 counts rooted trees by nodes and internals.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A034781, A055277, A206487, A358576, A358578, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],[_],{0,Infinity}],{n,100}] (* Gus Wiseman, Nov 28 2022 *)

A342507(n) = if( n==1, 0, my(f=factor(n)); 1+sum(k=1,matsize(f)[1],A342507(primepi(f[k,1]))*f[k,2]));

a(1)=0 and a(n) = A061775(n) - A109129(n) for n > 1.

A033185 Rooted tree triangle read by rows: a(n,k) = number of forests with n nodes and k rooted trees.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 6, 3, 1, 1, 20, 16, 7, 3, 1, 1, 48, 37, 18, 7, 3, 1, 1, 115, 96, 44, 19, 7, 3, 1, 1, 286, 239, 117, 46, 19, 7, 3, 1, 1, 719, 622, 299, 124, 47, 19, 7, 3, 1, 1, 1842, 1607, 793, 320, 126, 47, 19, 7, 3, 1, 1, 4766, 4235, 2095, 858, 327, 127, 47, 19, 7, 3, 1, 1

Offset: 1

Christian G. Bower

Leading column: A000081, rows sums: A000081 shifted.
Also, number of multigraphs of k components, n nodes, and no cycles except one loop in each component. See link below to have a picture showing the bijection between rooted forests and multigraphs of this kind. - Washington Bomfim, Sep 04 2010
Number of rooted trees with n+1 nodes and degree of the root is k.- Michael Somos, Aug 20 2018

			Triangle begins:
     1;
     1,    1;
     2,    1,   1;
     4,    3,   1,   1;
     9,    6,   3,   1,   1;
    20,   16,   7,   3,   1,  1;
    48,   37,  18,   7,   3,  1,  1;
   115,   96,  44,  19,   7,  3,  1,  1;
   286,  239, 117,  46,  19,  7,  3,  1,  1;
   719,  622, 299, 124,  47, 19,  7,  3,  1,  1;
  1842, 1607, 793, 320, 126, 47, 19,  7,  3,  1,  1;

Alois P. Heinz, Rows n = 1..141, flattened
W. Bomfim, Bijection between rooted forests and multigraphs without cycles except one loop in each connected component.
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Table 2.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000081, A005197, A106240, A181360, A027852 (2nd column), A000226 (3rd column), A029855 (4th column), A336087.

with(numtheory):
t:= proc(n) option remember; local d, j; `if` (n<=1, n,
      (add(add(d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
    end:
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j) *
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= (n, k)-> b(n, n, k):
seq(seq(a(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 20 2012

nn=10;f[x_]:=Sum[a[n]x^n,{n,0,nn}];sol=SolveAlways[0 == Series[f[x]-x Product[1/(1-x^i)^a[i],{i,1,nn}],{x,0,nn}],x];a[0]=0;g=Table[a[n],{n,1,nn}]/.sol//Flatten;h[list_]:=Select[list,#>0&];Map[h,Drop[CoefficientList[Series[x Product[1/(1-y x^i)^g[[i]],{i,1,nn}],{x,0,nn}],{x,y}],2]]//Grid  (* Geoffrey Critzer, Nov 17 2012 *)
t[1] = 1; t[n_] := t[n] = Module[{d, j}, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]; b[1, 1, 1] = 1; b[n_, i_, p_] := b[n, i, p] = If[p>n, 0, If[n == 0, 1, If[Min[i, p]<1, 0, Sum[b[n-i*j, i-1, p-j]*Binomial[t[i]+j-1, j], {j, 0, Min[n/i, p]}]]]]; a[n_, k_] := b[n, n, k]; Table[a[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

G.f.: 1/Product_{i>=1} (1-x*y^i)^A000081(i). - Vladeta Jovovic, Apr 28 2005

a(n, k) = sum over the partitions of n, 1M1 + 2M2 + ... + nMn, with exactly k parts, of Product_{i=1..n} binomial(A000081(i)+Mi-1, Mi). - Washington Bomfim, May 12 2005

A088957 Hyperbinomial transform of the sequence of 1's.

Original entry on oeis.org

1, 2, 6, 29, 212, 2117, 26830, 412015, 7433032, 154076201, 3608522954, 94238893883, 2715385121740, 85574061070045, 2928110179818478, 108110945014584623, 4284188833355367440, 181370804507130015569, 8169524599872649117330, 390114757072969964280163

Offset: 0

Paul D. Hanna, Oct 26 2003

nonn

See A088956 for the definition of the hyperbinomial transform.
a(n) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1.  The triangle A088956 classifies these functions according to the number of undefined elements in the domain.  The triangle A144289 classifies these functions according to the number of edges in their digraph representation (considering the empty function to have 1 edge).  The triangle A203092 classifies these functions according to the number of connected components. - Geoffrey Critzer, Dec 29 2011
a(n) is the number of rooted subtrees (for a fixed root) in the complete graph on n+1 vertices: a(3) = 29 is the number of rooted subtrees in K_4: 1 of size 1, 3 of size 2, 9 of size 3, and 16 spanning subtrees. - Alex Chin, Jul 25 2013 [corrected by Marko Riedel, Mar 31 2019]
From Gus Wiseman, Jan 28 2024: (Start)
Also the number of labeled loop-graphs on n vertices such that it is possible to choose a different vertex from each edge in exactly one way. For example, the a(3) = 29 uniquely choosable loop-graphs (loops shown as singletons) are:
  {}  {1}  {1,2}  {1,12}   {1,2,13}  {1,12,13}
      {2}  {1,3}  {1,13}   {1,2,23}  {1,12,23}
      {3}  {2,3}  {2,12}   {1,3,12}  {1,13,23}
                  {2,23}   {1,3,23}  {2,12,13}
                  {3,13}   {2,3,12}  {2,12,23}
                  {3,23}   {2,3,13}  {2,13,23}
                  {1,2,3}            {3,12,13}
                                     {3,12,23}
                                     {3,13,23}
(End)

			a(5) = 2117 = 1296 + 625 + 160 + 30 + 5 + 1 = sum of row 5 of triangle A088956.

Alois P. Heinz, Table of n, a(n) for n = 0..387
Marko Riedel et al., Proof of e.g.f. of sequence.

Cf. A088956 (triangle).
Row sums of A144289. - Alois P. Heinz, Jun 01 2009
Cf. A086331, A000169.
Column k=1 of A144303. - Alois P. Heinz, Oct 30 2012
The covering case is A000272, also the case of exactly n edges.
Without the choice condition we have A006125 (shifted left).
The unlabeled version is A087803.
The choosable version is A368927, covering A369140, loopless A133686.
The non-choosable version is A369141, covering A369142, loopless A367867.
Cf. A000081, A000085, A057500, A062740, A137916, A277473, A322661, A367904, A368596, A368597, A368924.

a088957 = sum . a088956_row  -- Reinhard Zumkeller, Jul 07 2013

a:= n-> add((n-j+1)^(n-j-1)*binomial(n,j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

nn = 16; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[Exp[x] Exp[t], {x, 0, nn}], x]  (* Geoffrey Critzer, Dec 29 2011 *)
With[{nmax = 50}, CoefficientList[Series[-LambertW[-x]*Exp[x]/x, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *)

x='x+O('x^10); Vec(serlaplace(-lambertw(-x)*exp(x)/x)) \\ G. C. Greubel, Nov 14 2017

a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k).
E.g.f.: A(x) = exp(x+sum(n>=1, n^(n-1)*x^n/n!)).
E.g.f.: -LambertW(-x)*exp(x)/x. - Vladeta Jovovic, Oct 27 2003
a(n) ~ exp(1+exp(-1))*n^(n-1). - Vaclav Kotesovec, Jul 08 2013
Binomial transform of A000272. - Gus Wiseman, Jan 25 2024

A206487 Number of ordered trees isomorphic (as rooted trees) to the rooted tree having Matula number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 4, 1, 3, 2, 6, 1, 1, 2, 2, 2, 6, 3, 2, 4, 4, 2, 6, 2, 3, 3, 2, 2, 5, 1, 3, 2, 6, 1, 4, 2, 4, 2, 4, 1, 12, 3, 2, 3, 1, 4, 6, 1, 3, 2, 6, 3, 10, 2, 6, 3, 3, 2, 12, 2, 5, 1, 4, 1, 12, 2, 4, 4, 4, 4, 12, 4, 3, 2, 4, 2, 6, 1, 3, 3, 6, 4, 6, 1, 8, 6, 2, 3, 10, 2, 6, 6, 5, 6, 6, 2, 6, 6, 2, 2, 20, 1, 6, 4, 3, 1, 12, 1, 1, 4, 12, 1, 12, 2, 2, 4, 4, 2, 6, 2, 12, 4, 6, 4, 15, 4, 4, 3, 9, 2, 12, 6, 4, 3, 6, 2, 24, 3, 4, 2, 6

Offset: 1

Emeric Deutsch, Apr 14 2012

nonn

The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

a(n) = the number of times n occurs in A127301. - Antti Karttunen, Jan 03 2013

			a(4)=1 because the rooted tree with Matula number 4 is V and there is no other ordered tree isomorphic to V. a(6)=2 because the rooted tree corresponding to n = 6 is obtained by joining the trees A - B and C - D - E at their roots A and C. Interchanging their order, we obtain another ordered tree, isomorphic (as rooted tree) to the first one.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
P. Schultz, Enumeration of rooted trees with an application to group presentations, Discrete Math., 41, 1982, 199-214.
Index entries for sequences related to Matula-Goebel numbers

Cf. A127301.
Positions of 1's are 1 and A214577.
Positions of first appearances are A358507, unsorted A358508.
A000108 counts ordered rooted trees, unordered A000081.
A061775 and A196050 count nodes and edges in Matula-Goebel trees.
Cf. A001263, A003238, A014486, A358506.

with(numtheory): F := proc (n) options operator, arrow: factorset(n) end proc: PD := proc (n) local k, m, j: for k to nops(F(n)) do m[k] := 0: for j while is(n/F(n)[k]^j, integer) = true do m[k] := m[k]+1 end do end do: [seq([F(n)[q], m[q]], q = 1 .. nops(F(n)))] end proc: a := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n)) else mul(a(PD(n)[j][1])^PD(n)[j][2], j = 1 .. nops(F(n)))*factorial(add(PD(n)[k][2], k = 1 .. nops(F(n))))/mul(factorial(PD(n)[k][2]), k = 1 .. nops(F(n))) end if end proc: seq(a(n), n = 1 .. 160);

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
MGTree[n_Integer]:=If[n===1,{},MGTree/@primeMS[n]]
treeperms[t_]:=Times@@Cases[t,b:{}:>Length[Permutations[b]],{0,Infinity}];
Table[treeperms[MGTree[n]],{n,100}] (* Gus Wiseman, Nov 21 2022 *)

a(1)=1; denoting by p(t) the t-th prime, if n = p(n_1)^{k_1}...p(n_r)^{k_r}, then a(n) = a(n_1)^{k_1}...a(n_r)^{k_r}*(k_1 + ... + k_r)!/[(k_1)!...(k_r)!] (see Theorem 1 in the Schultz reference, where the exponents k_j of N(n_j) have been inadvertently omitted).

A290760 Matula-Goebel numbers of transitive rooted identity trees (or transitive finitary sets).

Original entry on oeis.org

1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, 3390, 4290, 7878, 9570, 10230, 11310, 13026, 15510, 15990, 18330, 26070, 30966, 37290, 39390, 40890, 44070, 45210, 65130, 84810, 94830, 98310, 104610, 122070, 124410, 132990, 154830, 159330, 175890, 198330, 201630

Offset: 1

Gus Wiseman, Oct 19 2017

nonn

A rooted tree is transitive if every terminal subtree is a branch of the root. A finitary set is transitive if every element is also a subset.

			Let o = {}. The sequence of transitive finitary sets begins:
1     o
2     {o}
6     {o,{o}}
30    {o,{o},{{o}}}
78    {o,{o},{o,{o}}}
330   {o,{o},{{o}},{{{o}}}}
390   {o,{o},{{o}},{o,{o}}}
870   {o,{o},{{o}},{o,{{o}}}}
1410  {o,{o},{{o}},{{o},{{o}}}}
3198  {o,{o},{o,{o}},{{o,{o}}}}
3390  {o,{o},{{o}},{o,{o},{{o}}}}
4290  {o,{o},{{o}},{{{o}}},{o,{o}}}
7878  {o,{o},{o,{o}},{o,{o,{o}}}}
9570  {o,{o},{{o}},{{{o}}},{o,{{o}}}}
10230 {o,{o},{{o}},{{{o}}},{{{{o}}}}}
11310 {o,{o},{{o}},{o,{o}},{o,{{o}}}}
13026 {o,{o},{o,{o}},{{o},{o,{o}}}}
15510 {o,{o},{{o}},{{{o}}},{{o},{{o}}}}
15990 {o,{o},{{o}},{o,{o}},{{o,{o}}}}
18330 {o,{o},{{o}},{o,{o}},{{o},{{o}}}}

Cf. A000081, A001192, A004111, A007097, A076146, A276625, A279861, A290689, A290822.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
finitaryQ[n_]:=finitaryQ[n]=Or[n===1,With[{m=primeMS[n]},{UnsameQ@@m,finitaryQ/@m}]/.List->And];
subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
transitaryQ[n_]:=Divisible[n,Times@@subprimes[n]];
nn=100000;Fold[Select,Range[nn],{finitaryQ,transitaryQ}]

A301462 Number of enriched r-trees of size n.

Original entry on oeis.org

1, 2, 3, 8, 23, 77, 254, 921, 3249, 12133, 44937, 172329, 654895, 2565963, 9956885, 39536964, 156047622, 626262315, 2499486155, 10129445626, 40810378668, 166475139700, 676304156461, 2775117950448, 11342074888693, 46785595997544, 192244951610575, 796245213910406

Offset: 0

Gus Wiseman, Mar 21 2018

nonn

An enriched r-tree of size n > 0 is either a single node of size n, or a finite sequence of enriched r-trees with weakly decreasing sizes summing to n - 1.

These are different from the R-trees of data science and the enriched R-trees of Bousquet-Mélou and Courtiel.

			The a(3) = 8 enriched r-trees: 3, (2), ((1)), ((())), (11), (1()), (()1), (()()).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000081, A003238, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A290689, A300443, A301342-A301345, A301364-A301368, A301422, A301467, A301469, A301470.

ert[n_]:=ert[n]=1+Sum[Times@@ert/@y,{y,IntegerPartitions[n-1]}];
Array[ert,30]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

O.g.f.: 1/(1 - x) + x Product_{i > 0} 1/(1 - a(i) x^i).

A358371 Number of leaves in the n-th standard ordered rooted tree.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 13 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 25th ordered tree is (o,(o,o)) because the 24th composition is (1,4) and the 3rd composition is (1,1). Hence a(25) = 3.

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

The triangle counting trees by this statistic is A001263, unordered A055277.
The version for unordered trees is A109129, nodes A061775, edges A196050.
The nodes are counted by A358372.
A000081 counts unordered rooted trees, ranked by A358378.
A000108 counts ordered rooted trees.
A358374 ranks ordered identity trees, counted by A032027.
A358375 ranks ordered binary trees, counted by A126120
Cf. A004249, A005043, A063895, A187306, A284778, A358373, 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[Count[srt[n],{},{0,Infinity}],{n,100}]

A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 6, 7, 5, 16, 12, 12, 14, 10, 12, 9, 14, 19, 13, 10, 13, 17, 11, 32, 24, 24, 28, 20, 24, 18, 28, 38, 26, 20, 26, 34, 22, 24, 18, 18, 21, 15, 28, 21, 38, 53, 37, 26, 37, 43, 29, 20, 15, 26, 37, 23, 34, 43, 67, 41, 22, 29, 41, 59, 31, 64, 48, 48, 56, 40, 48, 36

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted general trees encoded in range [A014137(n-1)..A014138(n)] of A014486 to A000081(n+1) distinct non-oriented rooted general trees, encoded by their Matula-Goebel numbers. The latter encoding is explained in A061773.
A005517 and A005518 give the minimum and maximum value occurring in each such range.
Primes occur at positions given by A057548 (not in order, and with duplicates), and similarly, semiprimes, A001358, occur at positions given by A057518, and in general, A001222(a(n)) = A057515(n).
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127301(SP(n)) = A127301(n) for all n, then it preserves the non-oriented form of a general tree, which implies also that it is Łukasiewicz-word permuting, satisfying A129593(SP(n)) = A129593(n) for all n >= 0. Examples of such automorphisms include A072796, A057508, A057509/A057510, A057511/A057512, A057164, A127285/A127286 and A127287/A127288.
A206487(n) tells how many times n occurs in this sequence. - Antti Karttunen, Jan 03 2013

			A000081(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), encodes the following plane tree:
.....o
.....|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.
Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree:
.o
.|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(A000040(1)) * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two trees are identical if one ignores their orientation.

Antti Karttunen, Table of n, a(n) for n = 0..6917
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to Matula-Goebel numbers

a(A014138(n)) = A007097(n+1), a(A014137(n)) = A000079(n+1) for all n.
a(|A106191(n)|) = A033844(n-1) for all n >= 1.
Cf. A001222, A005517, A005518, A057515, A057518, A057548, A127302, A129593, A153826, A209638, A243491, A243492, A243494, A243496.
For standard instead of binary encoding we have A358506.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists binary encodings of ordered rooted trees.
Cf. A001263, A061775, A196050, A206487, A358505.

mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
binbalQ[n_]:=n==0||With[{dig=IntegerDigits[n,2]},And@@Table[If[k==Length[dig],SameQ,LessEqual][Count[Take[dig,k],0],Count[Take[dig,k],1]],{k,Length[dig]}]];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[mgnum[bint[n]],{n,Select[Range[0,1000],binbalQ]}] (* Gus Wiseman, Nov 22 2022 *)

(define (A127301 n) (*A127301 (A014486->parenthesization (A014486 n)))) ;; A014486->parenthesization given in A014486.
(define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))

A001222(a(n)) = A057515(n) for all n.

A244372 Number T(n,k) of unlabeled rooted trees with n nodes and maximal outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 2, 1, 0, 1, 10, 6, 2, 1, 0, 1, 22, 16, 6, 2, 1, 0, 1, 45, 43, 17, 6, 2, 1, 0, 1, 97, 113, 49, 17, 6, 2, 1, 0, 1, 206, 300, 136, 50, 17, 6, 2, 1, 0, 1, 450, 787, 386, 142, 50, 17, 6, 2, 1, 0, 1, 982, 2074, 1081, 409, 143, 50, 17, 6, 2, 1

Offset: 1

Joerg Arndt and Alois P. Heinz, Jun 26 2014

			The A000081(5) = 9 rooted trees with 5 nodes sorted by maximal outdegree are:
:  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  :                                 :             :         :
:     :                                 :             :         :
: -1- : ---------------2--------------- : -----3----- : ---4--- :
Thus row 5 = [0, 1, 5, 2, 1].
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,   1;
  0,  1,   2,    1;
  0,  1,   5,    2,    1;
  0,  1,  10,    6,    2,   1;
  0,  1,  22,   16,    6,   2,   1;
  0,  1,  45,   43,   17,   6,   2,  1;
  0,  1,  97,  113,   49,  17,   6,  2,  1;
  0,  1, 206,  300,  136,  50,  17,  6,  2,  1;
  0,  1, 450,  787,  386, 142,  50, 17,  6,  2,  1;
  0,  1, 982, 2074, 1081, 409, 143, 50, 17,  6,  2,  1;

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=2-10 give: A244398, A244399, A244400, A244401, A244402, A244403, A244404, A244405, A244406.
T(2n,n) gives A244407(n).
T(2n+1,n) gives A244410(n).
Row sum give A000081.
Cf. A244454.

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:
T:= (n, k)-> b(n-1$2, k$2) -`if`(k=0, 0, b(n-1$2, k-1$2)):
seq(seq(T(n, k), k=0..n-1), n=1..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]}]]]; T[n_, k_] := b[n-1, n-1, k, k] - If[k == 0, 0, b[n-1, n-1, k-1, k-1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 01 2014, translated from Maple *)

A301467 Number of enriched r-trees of size n with no empty subtrees.

Original entry on oeis.org

1, 2, 4, 8, 20, 48, 136, 360, 1040, 2944, 8704, 25280, 76320, 226720, 692992, 2096640, 6470016, 19799936, 61713152, 190683520, 598033152, 1863995392, 5879859200, 18438913536, 58464724992, 184356152832, 586898946048, 1859875518464, 5941384080384, 18901502482432

Offset: 1

Gus Wiseman, Mar 21 2018

nonn

An enriched r-tree of size n > 0 with no empty subtrees is either a single node of size n, or a finite nonempty sequence of enriched r-trees with no empty subtrees and with weakly decreasing sizes summing to n - 1.

			The a(4) = 8 enriched r-trees with no empty subtrees: 4, (3), (21), ((2)), (111), ((11)), ((1)1), (((1))).
The a(5) = 20 enriched r-trees with no empty subtrees:
  5,
  (4), ((3)), ((21)), (((2))), ((111)), (((11))), (((1)1)), ((((1)))),
  (31), (22), (2(1)), ((2)1), ((1)2), ((11)1), ((1)(1)), (((1))1),
  (211), ((1)11),
  (1111).

Alois P. Heinz, Table of n, a(n) for n = 1..1910

Cf. A000081, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A300660, A301342-A301345, A301364-A301368, A301422, A301462, A301469, A301470.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)* a(i)^j, j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, 1+b(n-1$2)):
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 21 2018

pert[n_]:=pert[n]=If[n===1,1,1+Sum[Times@@pert/@y,{y,IntegerPartitions[n-1]}]];
Array[pert,30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1] a[i]^j, {j, 0, n/i}]]];
a[n_] := a[n] = If[n < 2, n, 1 + b[n-1, n-1]];
Array[a, 30] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); v} \\ Andrew Howroyd, Aug 26 2018

O.g.f.: x^2/(1 - x) + x Product_{i > 0} 1/(1 - a(i) x^i).

cp's OEIS Frontend

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A033185 Rooted tree triangle read by rows: a(n,k) = number of forests with n nodes and k rooted trees.

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A088957 Hyperbinomial transform of the sequence of 1's.

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A206487 Number of ordered trees isomorphic (as rooted trees) to the rooted tree having Matula number n.

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A290760 Matula-Goebel numbers of transitive rooted identity trees (or transitive finitary sets).

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A301462 Number of enriched r-trees of size n.

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A358371 Number of leaves in the n-th standard ordered rooted tree.

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