

A184159


The difference between the levels of the highest and lowest leaves in the rooted tree with MatulaGoebel number n.


0



0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 3, 0, 1, 0, 2, 0, 1, 2, 2, 0, 0, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 0, 3, 0, 2, 1, 4, 0, 0, 1, 3, 0, 2, 1, 2, 2, 1, 0, 2, 1, 1, 2, 2, 3, 2, 0, 3, 0, 1, 0, 2, 2, 3, 1, 2, 1, 2, 3, 3, 1, 1, 0, 1, 2, 2, 2, 2, 0, 2, 1, 1, 1, 1, 2, 3
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OFFSET

1,10


COMMENTS

The MatulaGoebel number of a rooted tree is 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

Table of n, a(n) for n=1..110.
E. Deutsch, Tree statistics from Matula numbers, Arxiv preprint arXiv:1111.4288, 2011


FORMULA

In A184154 one constructs for each n the generating polynomial P(n,x) of the leaves of the rooted tree with MatulaGoebel number n, according to their levels. a(n) = degree of the numerator of P(n,1/x) (see the Maple program).


EXAMPLE

a(7)=0 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y with all leaves at level 2.
a(2^m)=0 because the rooted tree with MatulaGoebel number 2^m is the star with m edges; all leaves are at level 1.


MAPLE

with(numtheory): a := proc (n) local r, s, P: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: P := proc (n) 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: degree(numer(subs(x = 1/x, P(n)))) end proc; seq(a(n), n = 1 .. 110);


CROSSREFS

Cf. A184154
Sequence in context: A141684 A152492 A075446 * A231122 A178686 A142724
Adjacent sequences: A184156 A184157 A184158 * A184160 A184161 A184162


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 17 2011


STATUS

approved



