

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

Table of n, a(n) for n=2..116.


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

Cf. A109082.
Sequence in context: A038498 A319510 A257217 * A284441 A257992 A060952
Adjacent sequences: A184151 A184152 A184153 * A184155 A184156 A184157


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Oct 06 2011


EXTENSIONS

Keyword tabf added by Michel Marcus, Apr 09 2013


STATUS

approved



