

A184170


Number of vertices that have largest escape distance in the rooted tree having MatulaGoebel number n.


1



1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 3, 1, 3, 3, 2, 1, 4, 1, 1, 1, 1, 1, 3, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 1, 4, 1, 1
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OFFSET

1,6


COMMENTS

The escape distance of a vertex v in a rooted tree T is the distance from v to the nearest leaf of T that is a descendant of v. For the rooted tree ARBCDEF, rooted at R, the escape distance of B is 4 (the leaf A is not a descendant of B).
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.


REFERENCES

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.


LINKS



FORMULA

In A184167 one can find the generating polynomial P(n)=P(n,x) of the vertices of the rooted tree having MatulaGoebel number n, according to escape distance. a(n) is equal to the coefficient of the highest power of x.


EXAMPLE

a(7)=1 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y, having 4 vertices with escape distances 0,0,1, and 2; the largest value (2) occurs only once.
a(15)=2 because the rooted tree with Matula number 15 is the path tree ABRCDE, rooted at R; the escape distances are 0,1,2,2,1,0; the largest value (2) occurs twice.


MAPLE

with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))x^max(LLL(r(n)), LLL(s(n))) end if end proc: a := proc (n) options operator, arrow: coeff(P(n), x, degree(P(n))) end proc: seq(a(n), n = 1 .. 110);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



