A298426 -id:A298426 - cp's OEIS frontend

A184155 The Matula-Goebel number of rooted trees having all leaves at the same level.

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 21, 23, 25, 27, 31, 32, 49, 53, 57, 59, 63, 64, 67, 73, 81, 83, 85, 97, 103, 115, 121, 125, 127, 128, 131, 133, 147, 159, 171, 189, 227, 241, 243, 256, 269, 277, 289, 307, 311, 331, 335, 343, 361, 365, 367, 371, 391, 393, 399, 419, 425, 431, 439, 441, 477

Offset: 1

Emeric Deutsch, Oct 07 2011

nonn

The Matula-Goebel number of a rooted tree can be 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.

The sequence is infinite.

			7 is in the sequence because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having all leaves at level 2.
2^m is in the sequence for each positive integer m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
From _Gus Wiseman_, Mar 30 2018: (Start)
Sequence of trees begins:
01 o
02 (o)
03 ((o))
04 (oo)
05 (((o)))
07 ((oo))
08 (ooo)
09 ((o)(o))
11 ((((o))))
16 (oooo)
17 (((oo)))
19 ((ooo))
21 ((o)(oo))
23 (((o)(o)))
25 (((o))((o)))
27 ((o)(o)(o))
31 (((((o)))))
(End)

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 Review, 10, 1968, 273.

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A003238, A004111, A007097, A048816, A061775, A109082, A184154, A214577, A244925, A276625, A290689, A290760, A290822, A298422, A298424, A298426.

with(numtheory): P := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: A := {}: for n to 500 do if degree(numer(subs(x = 1/x, P(n)))) = 0 then A := `union`(A, {n}) else  end if end do: A;

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dep[n_]:=If[n===1,0,1+Max@@dep/@primeMS[n]];
rnkQ[n_]:=And[SameQ@@dep/@primeMS[n],And@@rnkQ/@primeMS[n]];
Select[Range[2000],rnkQ] (* Gus Wiseman, Mar 30 2018 *)

In A184154 one constructs for each n the generating polynomial P(n,x) of the leaves of the rooted tree with Matula-Goebel number n, according to their levels. The Maple program finds those n (between 1 and 500) for which P(n,x) is a monomial.

A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 4, 5, 3, 1, 3, 7, 12, 12, 6, 1, 3, 9, 19, 28, 25, 11, 1, 4, 14, 36, 65, 81, 63, 24, 1, 4, 16, 48, 107, 172, 193, 136, 47, 1, 5, 22, 75, 192, 369, 522, 522, 331, 103, 1, 5, 25, 96, 284, 643, 1108, 1420, 1292, 750, 214, 1, 6

Offset: 1

Gus Wiseman, Mar 19 2018

A binary enriched p-tree of weight n is either a single node of weight n, or an ordered pair of binary enriched p-trees with weakly decreasing weights summing to n.

			Triangle begins:
  1
  1   1
  1   1   1
  1   2   3   2
  1   2   4   5   3
  1   3   7  12  12   6
  1   3   9  19  28  25  11
  1   4  14  36  65  81  63  24
  1   4  16  48 107 172 193 136  47
  1   5  22  75 192 369 522 522 331 103
  ...
The T(6,3) = 7 binary enriched p-trees: ((41)1), ((32)1), (4(11)), ((31)2), ((22)2), (3(21)), ((21)3).

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

Last entries of each row give A000992. Row sums are A300443.

Cf. A001190, A008284, A055277, A063834, A196545, A273873, A289501, A292050, A298422, A298426, A300354, A300439, A300442, A301344, A301364-A301367.

bintrees[n_]:=Prepend[Join@@Table[Tuples[bintrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]===2&]}],n];
Table[Length[Select[bintrees[n],Count[#,_Integer,{-1}]===k&]],{n,13},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sum(k=1, n\2, v[k]*v[n-k])); apply(p->Vecrev(p/y), v)}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018

A301344 Regular triangle where T(n,k) is the number of semi-binary rooted trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 1, 0, 0, 1, 6, 4, 0, 0, 0, 1, 9, 11, 2, 0, 0, 0, 1, 12, 24, 9, 0, 0, 0, 0, 1, 16, 46, 32, 3, 0, 0, 0, 0, 1, 20, 80, 86, 20, 0, 0, 0, 0, 0, 1, 25, 130, 203, 86, 6, 0, 0, 0, 0, 0, 1, 30, 200, 423, 283, 46, 0, 0, 0, 0, 0, 0, 1, 36, 295, 816, 786, 234, 11, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Mar 19 2018

A rooted tree is semi-binary if all outdegrees 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.

			Triangle begins:
1
1   0
1   1   0
1   2   0   0
1   4   1   0   0
1   6   4   0   0   0
1   9  11   2   0   0   0
1  12  24   9   0   0   0   0
1  16  46  32   3   0   0   0   0
1  20  80  86  20   0   0   0   0   0
1  25 130 203  86   6   0   0   0   0   0
The T(6,3) = 4 semi-binary rooted trees: ((o(oo))), (o((oo))), (o(o(o))), ((o)(oo)).

Cf. A000081, A000598, A001190, A001678, A003238, A004111, A055277, A111299, A273873, A290689, A291636, A292050, A298118, A298204, A298422, A298426, A301342, A301343, A301345.

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

A298424 Matula-Goebel numbers of rooted trees in which all positive outdegrees are the same.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 14, 16, 31, 32, 49, 64, 76, 86, 127, 128, 256, 301, 424, 454, 512, 709, 722, 886, 1024, 1532, 1589, 1849, 2048, 2096, 3101, 3986, 4096, 5381, 6418, 6859, 8192, 9761, 9952, 11236, 13766, 13951, 14554, 16384, 19049, 21884, 22463, 23512

Offset: 1

Gus Wiseman, Jan 19 2018

nonn

			Sequence of trees begins:
1   o
2   (o)
3   ((o))
4   (oo)
5   (((o)))
8   (ooo)
11  ((((o))))
14  (o(oo))
16  (oooo)
31  (((((o)))))
32  (ooooo)
49  ((oo)(oo))
64  (oooooo)
76  (oo(ooo))
86  (o(o(oo)))
127 ((((((o))))))
128 (ooooooo)
256 (oooooooo)
301 ((oo)(o(oo)))
424 (ooo(oooo))
454 (o((oo)(oo)))
512 (ooooooooo)
709 (((((((o)))))))
722 (o(ooo)(ooo))
886 (o(o(o(oo))))

Cf. A000081, A000598, A001190, A003238, A007097, A111299, A214577, A276625, A298422, A298423, A298426.

MGtree[n_]:=If[n===1,{},MGtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
soQ[n_]:=Or[n===1,SameQ@@Length/@Cases[MGtree[n],{},{0,Infinity}]];
Select[Range[1000],soQ]

A301343 Regular triangle where T(n,k) is the number of planted achiral (or generalized Bethe) trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 3, 2, 2, 1, 1, 0, 1, 3, 2, 2, 1, 1, 1, 0, 1, 4, 2, 4, 1, 2, 1, 1, 0, 1, 4, 3, 4, 1, 3, 1, 1, 1, 0, 1, 5, 3, 6, 2, 4, 1, 2, 1, 1, 0, 1, 5, 3, 6, 2, 4, 1, 2, 1, 1, 1, 0, 1, 6, 4, 9, 2, 7, 1, 4, 2, 2, 1, 1, 0

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1  0
1  1  0
1  1  1  0
1  2  1  1  0
1  2  1  1  1  0
1  3  2  2  1  1  0
1  3  2  2  1  1  1  0
1  4  2  4  1  2  1  1  0
1  4  3  4  1  3  1  1  1  0
1  5  3  6  2  4  1  2  1  1  0
The T(9,4) = 4 planted achiral trees: (((((oooo))))), ((((oo)(oo)))), (((oo))((oo))), ((o)(o)(o)(o)).

Row sums are A003238. A version without the zeroes or first row is A214575.

Cf. A000081, A003238, A004111, A032305, A055277, A214577, A273873, A289078, A289079, A290689, A291443, A298422, A298426, A301342, A301344, A301345.

tri[n_,k_]:=If[k===1,1,If[k>=n,0,Sum[tri[n-k,d],{d,Divisors[k]}]]];
Table[tri[n,k],{n,10},{k,n}]

T(n,1) = 1, T(n,k) = 0 if n <= k, otherwise T(n,k) = Sum_{d|k} T(n - k, d).

A301367 Regular triangle where T(n,k) is the number of orderless same-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 3, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 4, 4, 3, 5, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 1, 2, 1, 1, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 4, 5, 10, 11, 14, 12, 14, 7, 13, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Mar 19 2018

An orderless same-tree of weight n > 0 is either a single node of weight n, or a finite multiset of two or more orderless same-trees whose weights are all the same and sum to n.

			Triangle begins:
1
1   1
1   0   1
1   1   1   2
1   0   0   0   1
1   1   1   2   1   3
1   0   0   0   0   0   1
1   1   1   3   4   4   3   5
1   0   1   0   1   0   1   0   2
1   1   0   0   1   2   1   1   1   3
1   0   0   0   0   0   0   0   0   0   1
1   1   2   4   5  10  11  14  12  14   7  13
1   0   0   0   0   0   0   0   0   0   0   0   1
1   1   0   0   0   0   1   2   1   1   1   1   1   3
The T(8,5) = 4 orderless same-trees: (4((11)(11))), (4(1111)), ((22)(2(11))), (222(11)).

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Last entries of each row give A289079. Row sums are A289078.

Cf. A003238, A006241, A008284, A055277, A063834, A273873, A281145, A289501, A294080, A298422, A298426, A299201, A299203, A301343, A301364-A301368.

olstrees[n_]:=Prepend[Join@@Table[Select[Tuples[olstrees/@ptn],OrderedQ],{ptn,Select[IntegerPartitions[n],Length[#]>1&&SameQ@@#&]}],n];
Table[Length[Select[olstrees[n],Count[#,_Integer,{-1}]===k&]],{n,14},{k,n}]

S(g, k)={polcoef(exp(sum(i=1, k, x^i*subst(g, y, y^i)/i) + O(x*x^k)), k)}
A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sumdiv(n, d, S(v[n/d], d))); apply(p -> Vecrev(p/y), v)}
{ my(v=A(16)); for(n=1, #v, print(v[n])) } \\ Andrew Howroyd, Aug 20 2018

A301366 Regular triangle where T(n,k) is the number of same-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 1, 1, 5, 3, 3, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 6, 12, 14, 12, 6, 1, 0, 1, 0, 3, 0, 3, 0, 2, 1, 1, 0, 0, 1, 7, 10, 10, 5, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 7, 21, 41, 58, 100, 100, 94, 48, 20

Offset: 1

Gus Wiseman, Mar 19 2018

A same-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more same-trees whose weights are all the same and sum to n.

			Triangle begins:
1
1   1
1   0   1
1   1   2   2
1   0   0   0   1
1   1   1   5   3   3
1   0   0   0   0   0   1
1   1   2   6  12  14  12   6
1   0   1   0   3   0   3   0   2
1   1   0   0   1   7  10  10   5   3
1   0   0   0   0   0   0   0   0   0   1
1   1   3   7  21  41  58 100 100  94  48  20
The T(8,4) = 6 same-trees: (4(2(11))), (4((11)2)), ((22)(22)), ((2(11))4), (((11)2)4), (2222).

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Last entries of each row give A006241. Row sums are A281145.

Cf. A003238, A008284, A055277, A063834, A273873, A289501, A294080, A298422, A298426, A299201, A299203, A300442, A300443, A301343, A301364-A301368.

sametrees[n_]:=Prepend[Join@@Table[Tuples[sametrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&&SameQ@@#&]}],n];
Table[Length[Select[sametrees[n],Count[#,_Integer,{-1}]===k&]],{n,12},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = x + sumdiv(n, d, v[n/d]^d)); apply(p -> Vecrev(p/x), v)}
{my(v=A(16)); for(n=1, #v, print(v[n]))} \\ Andrew Howroyd, Aug 20 2018

cp's OEIS Frontend

A184155 The Matula-Goebel number of rooted trees having all leaves at the same level.

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Maple

Mathematica

Formula

A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

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Mathematica

PARI

A301344 Regular triangle where T(n,k) is the number of semi-binary rooted trees with n nodes and k leaves.

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Programs

Mathematica

A298424 Matula-Goebel numbers of rooted trees in which all positive outdegrees are the same.

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Programs

Mathematica

A301343 Regular triangle where T(n,k) is the number of planted achiral (or generalized Bethe) trees with n nodes and k leaves.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A301367 Regular triangle where T(n,k) is the number of orderless same-trees of weight n with k leaves.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A301366 Regular triangle where T(n,k) is the number of same-trees of weight n with k leaves.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI