

A184167


Irregular triangle read by rows: T(n,k) is the number of vertices having escape distance k>=0 in the rooted tree having MatulaGoebel number n.


4



1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 4, 1, 2, 1, 1, 1, 3, 3, 3, 1, 1, 3, 2, 1, 3, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 2, 2, 2, 2, 1, 3, 3, 3, 3, 1, 4, 2, 2, 2, 2, 3, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 2, 2, 1, 3, 2, 1, 3, 2, 2, 4, 3, 3, 2, 1, 4, 2, 3, 3, 1, 4, 2, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,7


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.
Each row is nonincreasing (each vertex with escape distance k (k>=1) is the parent of some vertex with escape distance k1).


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..115.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to MatulaGoebel numbers


FORMULA

We give the recursive construction of the row generating polynomials P(n)=P(n,x): if n = p(t) (=the tth prime), then P(n)=P(t)+x^{1+LLL(t)}; if n=rs (r,s>=2), then P(n)=P(r)+P(s)x^{max(LLL(r),LLL(s))}; LLL denotes the level of the lowest leaf (computed recursively and programmed in A184166) (2nd Maple program yields P(n)).


EXAMPLE

Row n=7 is [2,1,1] because the rooted tree with MatulaGoebel number 7 is the rooted tree Y, having 2 leaves and 1 (1) vertex at distance 1 (2) from either of the leaves.
Triangle starts:
1;
1,1;
1,1,1;
2,1;
1,1,1,1;
2,2;
2,1,1;


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: for n to 30 do seq(coeff(P(n), x, k), k = 0 .. degree(P(n))) end do; # yields sequence in triangular form
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: P(998877665544);


CROSSREFS

Cf. A184170.
Sequence in context: A073454 A124765 A080356 * A036541 A176505 A338521
Adjacent sequences: A184164 A184165 A184166 * A184168 A184169 A184170


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Oct 23 2011


STATUS

approved



