A358505 -id:A358505 - cp's OEIS frontend

A014486 List of totally balanced sequences of 2n binary digits written in base 10. Binary expansion of each term contains n 0's and n 1's and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.

0, 2, 10, 12, 42, 44, 50, 52, 56, 170, 172, 178, 180, 184, 202, 204, 210, 212, 216, 226, 228, 232, 240, 682, 684, 690, 692, 696, 714, 716, 722, 724, 728, 738, 740, 744, 752, 810, 812, 818, 820, 824, 842, 844, 850, 852, 856, 866, 868, 872, 880, 906, 908, 914

Offset: 0

Wouter Meeussen

The binary Dyck-Language (A063171) in decimal representation.
These encode width 2n mountain ranges, rooted planar trees of n+1 vertices and n edges, planar planted trees with n nodes, rooted plane binary trees with n+1 leaves (2n edges, 2n+1 vertices, n internal nodes, the root included), Dyck words, binary bracketings, parenthesizations, non-crossing handshakes and partitions and many other combinatorial structures in Catalan family, enumerated by A000108.
Is Sum_{k=1..n} a(k) / n^(5/2) bounded? - Benoit Cloitre, Aug 18 2002
This list is the intersection of A061854 and A031443. - Jason Kimberley, Jan 18 2013
The sequence does start at n = 0, since in the binary interpretation of the Dyck language (e.g., as parenthesizations where "1" stands for "(" and "0" stands for ")") having a(0) = 0 will do since it would stand for the empty string where the "0"s and "1"s are balanced (hence the parentheses are balanced). - Daniel Forgues, Feb 17 2013
It appears that for n>=1 this sequence can be obtained by concatenating the terms of the irregular array whose n-th row length is A000108(n) and that is defined recursively by B(n,0) = A020988(n) and B(n,k) = B(n, k-1) + D(n, k-1) where D(x,y) = (2^(2*(A089309(B(x,y))-1))-1)*(2/3) + 2^A007814(B(x,y)). - Raúl Mario Torres Silva and Michel Marcus, May 01 2020
This encoding is related to the ranking by standard ordered tree numbers in that (1) the binary encoding of trees ordered by standard ranking is given by A358505, while (2) the standard ranking of trees ordered by binary encoding is given by A358523. - Gus Wiseman, Nov 21 2022

			a(19) = 226_10 = 11100010_2 = A063171(19) as bracket expression: ( ( ( ) ) )( ) and as a binary tree, proceeding from left to right in depth-first fashion, with 1's in binary expansion standing for internal (branching) nodes and 0's for leaves:
  0   0
   \ /
    1   0 0  (0)
     \ /   \ /
      1     1
       \   /
         1
Note that in this coding scheme the last leaf of the binary trees (here in parentheses) is implicit. This tree can be also converted to a particular S-expression in languages like Lisp, Scheme and Prolog, if we interpret its internal nodes (1's) as cons cells with each leftward leaning branch being the "car" and the rightward leaning branch the "cdr" part of the pair, with the terminal nodes (0's) being ()'s (NILs). Thus we have (cons (cons (cons () ()) ()) (cons () ())) = '( ( ( () . () ) . () ) . ( () . () ) ) = (((())) ()) i.e., the same bracket expression as above, but surrounded by extra parentheses. This mapping is performed by the Scheme function A014486->parenthesization given below.
From _Gus Wiseman_, Nov 21 2022: (Start)
The terms and corresponding ordered rooted trees begin:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))
(End)

Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 443 (Algorithm P).

Paolo Xausa, Table of n, a(n) for n = 0..23713 (terms 0..2500 from Franklin T. Adams-Watters).
Jason Bell, Thomas Finn Lidbetter, and Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
N. G. De Bruijn and B. J. M. Morselt, A note on plane trees, J. Combinatorial Theory 2 (1967), 27-34.
Gennady Eremin, Dynamics of balanced parentheses, lexicographic series and Dyck polynomials, arXiv:1909.07675 [math.CO], 2019.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
Antti Karttunen, Catalan ranking and unranking functions, OEIS Wiki.
Antti Karttunen, Illustration of 626 initial terms (up to size n=7) with various combinatorial interpretations of Catalan numbers encoded by this sequence.
Antti Karttunen, a089408.c - C program for computing this sequence and many of the related automorphisms.
Antti Karttunen, Some notes on Catalan's Triangle.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.
D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, Generation, Enumeration and Search, CRC Press, 1998.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
OEIS Wiki, Combinatorial interpretations of Catalan numbers
Frank Ruskey, Algorithmic Solution of Two Combinatorial Problems.
R. P. Stanley, Hipparchus, Plutarch, Schroeder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
R. P. Stanley, Exercises on Catalan and Related Numbers.
Paolo Xausa, A Mathematica implementation of Knuth's algorithms P and U.
Index entries for encodings of plane rooted trees (various subsets of this sequence).
Index entries for sequences related to parenthesizing
Index entries for signature-permutations induced by Catalan automorphisms (permutations of natural numbers induced by various bijective operations acting on these structures)
Index entries for the sequences induced by list functions of Lisp (sequences induced by various other operations on these codes or the corresponding structures).

Characteristic function: A080116. Inverse function: A080300.
The terms of binary width 2n are counted by A000108(n). Subset of A036990. Number of peaks in each mountain (number of leaves in rooted plane general trees): A057514. Number of trailing zeros in the binary expansion: A080237. First differences: A085192.
Cf. also A009766, A014137, A071156, A079436, A085184, A213704.
Cf. A020988, A089309, A007814.
Branches of the ordered tree are counted by A057515.
Edges of the ordered tree are counted by A072643.
The Matula-Goebel number of the ordered tree is A127301.
The standard ranking of the ordered tree is A358523.
The depth of the ordered tree is A358550.
Nodes of the ordered tree are counted by A358551.
Cf. A000081, A001263, A358505, A358524.

# Maple procedure CatalanUnrank is adapted from the algorithm 3.24 of the CAGES book and the Scheme function CatalanUnrank from Ruskey's thesis. See the a089408.c program for the corresponding C procedures.
CatalanSequences := proc(upto_n) local n,a,r; a := []; for n from 0 to upto_n do for r from 0 to (binomial(2*n,n)/(n+1))-1 do a := [op(a),CatalanUnrank(n,r)]; od; od; return a; end;
CatalanUnrank := proc(n,rr) local r,x,y,lo,m,a; r := (binomial(2*n,n)/(n+1))-(rr+1); y := 0; lo := 0; a := 0; for x from 1 to 2*n do m := Mn(n,x,y+1); if(r <= lo+m-1) then y := y+1; a := 2*a + 1; else lo := lo+m; y := y-1; a := 2*a; fi; od; return a; end;
Mn := (n,x,y) -> binomial(2*n-x,n-((x+y)/2)) - binomial(2*n-x,n-1-((x+y)/2));
# Alternative:
bin := n -> ListTools:-Reverse(convert(n, base, 2)):
isA014486 := proc(n): local B, s, b; s := 0;
    if n > 0 then
      for b in bin(n) do
          s := s + ifelse(b = 1, 1, -1);
           if 0 > s then return false fi;
      od fi;
  s = 0 end:
select(isA014486, [seq(0..240)]);  # Peter Luschny, Mar 13 2024

cat[ n_ ] := (2 n)!/n!/(n+1)!; b2d[li_List] := Fold[2#1+#2&, 0, li]
d2b[n_Integer] := IntegerDigits[n, 2]
tree[n_] := Join[Table[1, {i, 1, n}], Table[0, {i, 1, n}]]
nexttree[t_] := Flatten[Reverse[t]/. {a___, 0, 0, 1, b___}:> Reverse[{Sort[{a, 0}]//Reverse, 1, 0, b}]]
wood[ n_ /; n<8 ] := NestList[ nexttree, tree[ n ], cat[ n ]-1 ]
Table[ Reverse[ b2d/@wood[ j ] ], {j, 0, 6} ]//Flatten
(* Alternative code *)
tbQ[n_]:=Module[{idn2=IntegerDigits[n,2]},Count[idn2,1]==Length[idn2]/2&&Min[Accumulate[idn2/.{0->-1}]]>=0]; Join[{0},Select[Range[900],tbQ]] (* Harvey P. Dale, Jul 04 2013 *)
balancedQ[0] = True; balancedQ[n_] := Module[{s = 0}, Do[s += If[b == 1, 1, -1]; If[s < 0, Return[False]], {b, IntegerDigits[n, 2]}]; Return[s == 0] ]; A014486 = FromDigits /@ IntegerDigits[Select[Range[0, 1000], balancedQ ]] (* Jean-François Alcover, Mar 05 2016 *)
A014486Q[0] = True; A014486Q[n_] := Catch[Fold[If[# < 0, Throw[False], If[#2 == 0, # - 1, # + 1]] &, 0, IntegerDigits[n, 2]] == 0]; Select[Range[0, 880], A014486Q] (* JungHwan Min, Dec 11 2016 *)
(* Uses Algorithm P from Knuth's TAOCP section 7.2.1.6 - see References and Links. *)
alist[n_] := Block[{a = Flatten[Table[{1, 0}, n]], m = 2*n - 1, j, k},
    FromDigits[#, 2]& /@ Reap[
    While[True,
        Sow[a]; a[[m]] = 0;
        If[a[[m - 1]] == 0,
            a[[--m]] = 1, j = m - 1; k = 2*n - 1;
            While[j > 1 && a[[j]] == 1, a[[j--]] = 0; a[[k]] = 1; k -= 2];
            If[j == 1, Break[]];
            a[[j]] = 1; m = 2*n - 1]
    ]][[2, 1]]];
Join[{{0}, {2}}, Array[alist, 4, 2]] (* Paolo Xausa, Mar 16 2024 *)

isA014486(n)=my(v=binary(n),t=0);for(i=1,#v,t+=if(v[i],1,-1);if(t<0,return(0))); t==0 \\ Charles R Greathouse IV, Jun 10 2011

a_rows(N) = my(a=Vec([[0]], N)); for(r=1, N-1, my(b=a[r], c=List()); foreach(b, t, my(v=if(t, valuation(t, 2), 0)); for(i=0, v, listput(~c, (t<<2)+(2<Ruud H.G. van Tol, May 16 2024

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A014486_gen(): # generator of terms
    yield 0
    for l in count(1):
        for s in multiset_permutations('0'*l+'1'*(l-1)):
            c, m = 0, (l<<1)-1
            for i in range(m):
                if s[i] == '1':
                    c += 2
                if cA014486_list = list(islice(A014486_gen(),30)) # Chai Wah Wu, Nov 28 2023

def is_A014486(n) :
    B = bin(n)[2::] if n != 0 else 0
    s = 0
    for b in B :
        s += 1 if b=='1' else -1
        if 0 > s : return False
    return 0 == s
def A014486_list(n): return [k for k in (1..n) if is_A014486(k) ]
A014486_list(888) # Peter Luschny, Aug 10 2012

Additional comments from Antti Karttunen, Aug 11 2000 and May 25 2004

Added a(0)=0 (which had been removed in June 2011), Joerg Arndt, Feb 27 2013

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.

A358506 Matula-Goebel number of the n-th standard ordered rooted tree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 8, 7, 10, 9, 12, 10, 12, 12, 16, 11, 14, 15, 20, 15, 18, 18, 24, 14, 20, 18, 24, 20, 24, 24, 32, 13, 22, 21, 28, 25, 30, 30, 40, 21, 30, 27, 36, 30, 36, 36, 48, 22, 28, 30, 40, 30, 36, 36, 48, 28, 40, 36, 48, 40, 48, 48, 64, 13, 26, 33, 44

Offset: 1

Gus Wiseman, Nov 20 2022

nonn

First differs from A333219 at a(65) = 13, A333219(65) = 17.
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.
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 first eight standard ordered trees are: o, (o), ((o)), (oo), (((o))), ((o)o), (o(o)), (ooo), with Matula-Goebel numbers: 1, 2, 3, 4, 5, 6, 6, 8.

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

For binary instead of standard encoding we have A127301.
There are exactly A206487(n) appearances of n.
For binary instead of Matula-Goebel encoding we have A358505.
Positions of first appearances are A358522, sorted A358521.
A000108 counts ordered rooted trees, unordered A000081.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A014486, A061775, A196050, A358371, A358372, A358378, A358379, A358507, A358508.

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

A358523 Standard ordered tree numbers of ordered trees in order of their binary encodings (A014486).

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 6, 9, 5, 16, 15, 14, 25, 13, 12, 11, 18, 129, 65, 10, 33, 257, 17, 32, 31, 30, 57, 29, 28, 27, 50, 385, 193, 26, 97, 769, 49, 24, 23, 22, 41, 21, 36, 35, 258, 32769, 16385, 130, 8193, 16777217, 4097, 20, 19, 66

Offset: 0

Gus Wiseman, Nov 21 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 binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

			The first six binary encodings are: 0, 2, 10, 12, 42, 44, and the corresponding trees have standard ranks: 1, 2, 4, 3, 8, 7.

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

A dual sequence is A358505.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists all binary encodings.
Cf. A001263, A014221, A057122, A358371, A358372, A358373, A358379.

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
srtinv[t_]:=If[t=={},1,stcinv[srtinv/@t]+1];
binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]]
Table[srtinv[bint[n]],{n,Select[Range[0,100],binbalQ]}]

A358521 Sorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 16, 17, 18, 19, 20, 22, 24, 32, 33, 34, 35, 36, 37, 38, 40, 43, 44, 48, 64, 66, 67, 68, 69, 70, 72, 74, 75, 76, 80, 86, 88, 96, 128, 129, 131, 132, 133, 134, 136, 137, 138, 139, 140, 144, 147, 148, 150, 152, 160, 171, 172

Offset: 1

Gus Wiseman, Nov 20 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 standard ordered trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: ((o)o)
   8: (ooo)
   9: ((oo))
  10: (((o))o)
  11: ((o)(o))
  12: ((o)oo)
  16: (oooo)
  17: ((((o))))
  18: ((oo)o)
  19: (((o))(o))
  20: (((o))oo)

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

Positions of first appearances in A358506.
The unsorted version is A358522.
A000108 counts ordered rooted trees, unordered A000081.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
fir[Table[mgnum[srt[n]],{n,100}]]

A358522 Least number k such that the k-th standard ordered tree has Matula-Goebel number n, i.e., A358506(k) = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 11, 10, 17, 12, 33, 18, 19, 16, 257, 22, 129, 20, 35, 34, 1025, 24, 37, 66, 43, 36, 513, 38, 65537, 32, 67, 514, 69, 44, 2049, 258, 131, 40

Offset: 1

Gus Wiseman, Nov 20 2022

			The terms together with their standard ordered trees begin:
    1: o
    2: (o)
    3: ((o))
    4: (oo)
    5: (((o)))
    6: ((o)o)
    9: ((oo))
    8: (ooo)
   11: ((o)(o))
   10: (((o))o)
   17: ((((o))))
   12: ((o)oo)
   33: (((o)o))
   18: ((oo)o)
   19: (((o))(o))
   16: (oooo)
  257: (((oo)))
   22: ((o)(o)o)
  129: ((ooo))
   20: (((o))oo)
   35: ((oo)(o))
   34: ((((o)))o)

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

Position of first appearance of n in A358506.
The sorted version is A358521.
A000108 counts ordered rooted trees, unordered A000081.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
uv=Table[mgnum[srt[n]],{n,10000}];
Table[Position[uv,k][[1,1]],{k,Min@@Complement[Range[Max@@uv],uv]-1}]

cp's OEIS Frontend

A014486 List of totally balanced sequences of 2n binary digits written in base 10. Binary expansion of each term contains n 0's and n 1's and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.

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

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A358506 Matula-Goebel number of the n-th standard ordered rooted tree.

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A358523 Standard ordered tree numbers of ordered trees in order of their binary encodings (A014486).

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A358521 Sorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).

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A358522 Least number k such that the k-th standard ordered tree has Matula-Goebel number n, i.e., A358506(k) = n.

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A358550 Depth of the ordered rooted tree with binary encoding A014486(n).

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A358551 Number of nodes in the ordered rooted tree with binary encoding A014486(n).