A191400 Number of nonroot vertices of degree 2 in the rooted tree having Matula-Goebel number n.

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

Offset

1,5

Comments

The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel 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 Matula-Goebel numbers of the m branches of T.

References

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

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 Matula-Goebel numbers

Formula

Let g(n)=G(n,x) be the generating polynomial of the nonroot nodes of degree 2 of the rooted tree having Matula-Goebel number n, with respect to level. Then g(1)=g(2)=0; if n = p(t) (=the t-th prime) and t is prime, then g(n)=x+x*g(t); if n=p(t) (=the t-th 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 Matula-Goebel number 5 is the path tree on 4 vertices. a(7)=0 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, with no vertices of degree 2.

Crossrefs

Sequence in context: A327757 A373122 A373923 * A168315 A120730 A122851

Adjacent sequences: A191397 A191398 A191399 * A191401 A191402 A191403

Keyword

nonn

Author

Emeric Deutsch, Dec 10 2011

Status

approved