A198323 Matula-Goebel number of rooted trees that have only vertices of degree 1 and of maximal degree.

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

Author

Emeric Deutsch, Oct 29 2011

Keywords

• nonn

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.

Examples

			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.
		

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
• D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A182907.

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.