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

%I A198323 #15 Mar 07 2017 06:16:27
%S A198323 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,
%T A198323 98,121,127,128,131,152,155,172,227,254,256,311,341,343,381,383,443,
%U A198323 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
%N A198323 Matula-Goebel number of rooted trees that have only vertices of degree 1 and of maximal degree.
%C A198323 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.
%H A198323 M. Fischermann, A. Hoffmann, D. Rautenbach, L. Székely, and L. Volkmann, <a href="http://dx.doi.org/10.1016/S0166-218X(01)00357-2">Wiener index versus maximum degree in trees</a>, Discrete Applied. Math., 122, 2002, 127-137.
%H A198323 F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A198323 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A198323 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22
%H A198323 D. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A198323 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A198323 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.
%e A198323 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.
%e A198323 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.
%Y A198323 Cf. A182907.
%K A198323 nonn
%O A198323 1,2
%A A198323 _Emeric Deutsch_, Oct 29 2011