A198336 Irregular triangle read by rows: row n is the sequence of Matula-Goebel numbers of the rooted trees obtained from the rooted tree with Matula-Goebel 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, 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

Offset: 1

Author

Emeric Deutsch, Dec 01 2011

Keywords

• nonn
• tabf

Comments

This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini - Consonni reference (p. 42).

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.

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.

Examples

			Row 7 is 7, 1 because the rooted tree with Matula-Goebel number 7 is Y and after the first pruning we obtain the 1-vertex tree having Matula-Goebel number 1. Row 5 is 5, 2, 1 because it refers to the path tree on four vertices; after pruning it becomes the 1-edge tree with Matula-Goebel 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;
		

References

• A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
• 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.
• R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

Crossrefs

Cf. A198329.

Programs

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;

Formula

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