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A198323
Matula-Goebel number of rooted trees that have only vertices of degree 1 and of maximal degree.
0
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 16, 19, 22, 25, 28, 31, 32, 33, 43, 53, 55, 62, 64, 93, 98, 121, 127, 128, 131, 152, 155, 172, 227, 254, 256, 311, 341, 343, 381, 383, 443, 512, 602, 635, 709, 719, 848, 908, 961, 1024, 1397, 1418, 1444, 1619, 1772, 1993, 2048, 2107, 2127, 2939, 3064, 3178, 3209, 3545, 3671, 3698, 3937
OFFSET
1,2
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.
LINKS
M. Fischermann, A. Hoffmann, D. Rautenbach, L. Székely, and L. Volkmann, Wiener index versus maximum degree in trees, Discrete Applied. Math., 122, 2002, 127-137.
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
FORMULA
In A182907 one can find the generating polynomial g(n,x) of the vertices of the rooted tree having Matula-Goebel number n, according to degree. We look for those values of n for which the polynomial g(n) = g(n,x) has at most 2 terms.
EXAMPLE
7 is in the sequence because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 3 vertices of degree 1 and 1 vertex of degree 3.
15, 22, and 31 are in the sequence because the corresponding rooted trees are path trees on 6 vertices (with different roots); they have 2 vertices of degree 1 and 4 vertices of degree 2.
CROSSREFS
Cf. A182907.
Sequence in context: A194975 A299441 A069751 * A254649 A261293 A177872
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 29 2011
STATUS
approved