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A191400 Number of nonroot vertices of degree 2 in the rooted tree having Matula-Goebel 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 (list; graph; refs; listen; history; text; internal format)
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: A122856 A055791 A245842 * A168315 A120730 A122851

Adjacent sequences:  A191397 A191398 A191399 * A191401 A191402 A191403

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 10 2011

STATUS

approved

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Last modified February 25 21:58 EST 2018. Contains 299660 sequences. (Running on oeis4.)