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 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.
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.
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.
EXAMPLE
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;
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
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 01 2011
STATUS
approved