

A184154


Triangle read by rows: T(n,k) is the number of leaves at level k>=1 of the rooted tree having MatulaGoebel number n (n>=2).


2



1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 2, 3, 0, 2, 1, 0, 1, 0, 0, 0, 1, 2, 1, 0, 1, 1, 1, 2, 0, 1, 1, 4, 0, 0, 2, 1, 2, 0, 3, 2, 0, 1, 0, 3, 1, 0, 0, 1, 0, 0, 2, 3, 1, 0, 0, 2, 1, 1, 1, 0, 3, 2, 2, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 2, 1, 2, 2, 0, 2, 1, 1, 3, 0, 2, 1, 3, 0, 1, 0, 0, 1, 1, 1, 3, 0, 1, 2, 2, 0, 0, 1, 0, 2, 1
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OFFSET

2,4


COMMENTS

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.
Number of entries in row n is A109082(n) (n=2,3,...).


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

We give the recursive construction of the row generating polynomials P(n)=P(n,x). P(2)=x; if n = p(t) (=the tth prime), then P(n)=x*P(t); if n=rs (r,s>=2), then P(n)=P(r)+P(s) (2nd Maple program yields P(n)).


EXAMPLE

Row n=7 is [0,2] because the rooted tree with MatulaGoebel number 7 is the rooted tree Y, having 0 leaves at level 1 and 2 leaves at level 2.
Row n=2^m is [m] because the rooted tree with MatulaGoebel number 2^m is a star with m edges.
Triangle starts:
1;
0,1;
2;
0,0,1;
1,1;
0,2;
3;
0,2;


MAPLE

with(numtheory): P := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 2 then x elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: for n from 2 to 30 do seq(coeff(P(n), x, k), k = 1 .. degree(P(n))) end do; # yields sequence in triangular form
with(numtheory): P := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 2 then x elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: for n from 2 to 30 do P(n) end do;


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



EXTENSIONS



STATUS

approved



