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%I #11 Mar 07 2017 11:31:25
%S 1,0,1,1,2,2,0,0,3,3,3,1,1,1,4,1,1,2,1,2,2,4,2,2,5,2,3,0,2,3,4,0,5,2,
%T 3,3,2,2,3,3,2,1,0,3,4,3,3,1,1,4,3,1,0,4,6,1,3,3,2,4,3,5,2,1,4,4,2,1,
%U 4,2,3,2,1,3,5,1,4,2,3,2,5,3,3,2,4,1
%N Number of vertices of even degree in the rooted tree with 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.
%C The degree sequences of the rooted trees with Matula-Goebel number n are given in A182907.
%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F For a graph with degree sequence a,b,c,..., define the degree sequence polynomial to be x^a + x^b + x^c + ... . The degree sequence polynomial g(n)=g(n,x) of the rooted tree with Matula-Goebel number n can be obtained recursively in the following way: g(1)=1; if n=p(t) (=the t-th prime), then g(n)=g(t)+x^G(t)*(x-1)+x; if n=rs (r,s>=2), then g(n)=g(r)+g(s)-x^G(r)-x^G(s)+x^G(n); G(m) is the number of prime divisors of m counted with multiplicities. Clearly, a(n)=(1/2)*(g(n,1) + g(n,-1)).
%e a(5)=2 because the rooted tree with Matula-Goebel number 5 is the path tree on 4 vertices and the vertex degrees are 1,1,2,2;
%e a(7)=0 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y having vertices of degree 1,1,1,3.
%p with(numtheory): g := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then sort(expand(g(pi(n))+x^bigomega(pi(n))*(x-1)+x)) else sort(expand(g(r(n))+g(s(n))-x^bigomega(r(n))-x^bigomega(s(n))+x^bigomega(n))) end if end proc: a := proc (n) options operator, arrow: (1/2)*subs(x = 1, g(n))+(1/2)*subs(x = -1, g(n)) end proc: seq(a(n), n = 1 .. 110);
%Y Cf. A182907, A190175.
%K nonn
%O 1,5
%A _Emeric Deutsch_, Dec 09 2011
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