

A198336


Irregular triangle read by rows: row n is the sequence of MatulaGoebel numbers of the rooted trees obtained from the rooted tree with MatulaGoebel number n by pruning it successively 0,1,2,... times. The operation of pruning consists of the removal of the vertices of degree one, together with their incident vertices.


1



1, 2, 1, 3, 1, 4, 1, 5, 2, 1, 6, 2, 1, 7, 1, 8, 1, 9, 4, 1, 10, 3, 1, 11, 3, 1, 12, 2, 1, 13, 2, 1, 14, 2, 1, 15, 6, 2, 1, 16, 1, 17, 2, 1, 18, 4, 1, 19, 1, 20, 3, 1, 21, 4, 1, 22, 5, 2, 1, 23, 4, 1, 24, 2, 1, 25, 9, 4, 1, 26, 3, 1, 27, 8, 1, 28, 2, 1, 29, 3, 1
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OFFSET

1,2


COMMENTS

This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini  Consonni reference (p. 42).
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.
First entry of row n is n, last entry is 1.
Number of entries in row n is 1 + the radius of the corresponding rooted tree.


REFERENCES

A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355375, 1979.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, WileyVCH, 2000.


LINKS

Table of n, a(n) for n=1..81.


FORMULA

A198329(n) is the MatulaGoebel number of the rooted tree obtained by removing from the rooted tree with MatulaGoebel number n the vertices of degree one, together with their incident edges. Repeated application of this yields the MatulaGoebel numbers of the trees obtained by successive prunings. The Maple program is based on this.


EXAMPLE

Row 7 is 7, 1 because the rooted tree with MatulaGoebel number 7 is Y and after the first pruning we obtain the 1vertex tree having MatulaGoebel number 1. Row 5 is 5, 2, 1 because it refers to the path tree on four vertices; after pruning it becomes the 1edge tree with MatulaGoebel number 2.
Triangle starts:
1;
2,1;
3,1;
4,1;
5,2,1;
6,2,1;
7,1;
8,1;
9,4,1;
10,3,1;


MAPLE

with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: S := proc (m) local A, i: A[m, 1] := m: for i while a(A[m, i]) < A[m, i] do A[m, i+1] := a(A[m, i]) end do: seq(A[m, j], j = 1 .. i) end proc: for n to 15 do S(n) end do;


CROSSREFS

Cf. A198329.
Sequence in context: A229994 A165818 A194746 * A290980 A007381 A308059
Adjacent sequences: A198333 A198334 A198335 * A198337 A198338 A198339


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Dec 01 2011


STATUS

approved



