

A191400


Number of nonroot vertices of degree 2 in the rooted tree having MatulaGoebel number n.


0



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

1,5


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.


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..86.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Let g(n)=G(n,x) be the generating polynomial of the nonroot nodes of degree 2 of the rooted tree having MatulaGoebel number n, with respect to level. Then g(1)=g(2)=0; if n = p(t) (=the tth prime) and t is prime, then g(n)=x+x*g(t); if n=p(t) (=the tth prime) and t is not prime, then g(n)=x*g(t); if n=rs (r,s>=2), then g(n)=g(r)+g(s). Clearly, a(n)=G(n,1).


EXAMPLE

a(5)=2 because the rooted tree with MatulaGoebel number 5 is the path tree on 4 vertices. a(7)=0 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y, with no vertices of degree 2.


CROSSREFS

Sequence in context: A122856 A055791 A245842 * A168315 A120730 A122851
Adjacent sequences: A191397 A191398 A191399 * A191401 A191402 A191403


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 10 2011


STATUS

approved



