A191515 - OEIS

%I #19 Nov 01 2021 12:44:30

%S 0,0,0,1,0,1,1,1,1,1,0,1,1,2,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,1,0,1,1,2,

%T 2,1,1,2,2,1,1,2,2,1,1,2,1,1,3,1,2,2,1,1,1,2,2,2,1,1,1,1,2,1,2,1,1,2,

%U 2,2,1,1,2,2,1,2,2,2,1,1,1,2,1,2,2,3

%N Number of vertices of outdegree >=2 in the rooted tree having Matula-Goebel number n.

%C 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 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.

%H 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 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 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 D. Matula, <a href="http://www.jstor.org/stable/2027327?seq=30">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F Let g(n)=G(n,x) be the generating polynomial of the vertices of outdegree >=2 of the rooted tree having Matula-Goebel number n, with respect to level. Then g(1)=0; if n = p(t) (=the t-th prime), then g(n)=x*g(t); if n=rs (r,s>=2), then g(n)=1+g(r)+g(s)-G(r,0)-G(s,0). Clearly, a(n)=G(n,1).

%e a(5)=0 because the rooted tree with Matula-Goebel number 5 is the path-tree on 4 vertices.

%e a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.

%o (PARI) a(n) = my(f=factor(n)); (vecsum(f[,2])>=2) + [self()(primepi(p))|p<-f[,1]]*f[,2]; \\ _Kevin Ryde_, Oct 30 2021

%Y Cf. A007097 (indices of 0's).

%K nonn

%O 1,14

%A _Emeric Deutsch_, Dec 10 2011