

A190174


Number of vertices of even degree in the rooted tree with MatulaGoebel number n.


1



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, 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, 4, 2, 3, 2, 1, 3, 5, 1, 4, 2, 3, 2, 5, 3, 3, 2, 4, 1
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OFFSET

1,5


COMMENTS

The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.
The degree sequences of the rooted trees with MatulaGoebel number n are given in A182907.


REFERENCES

F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.


LINKS

Table of n, a(n) for n=1..86.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288
Index entries for sequences related to MatulaGoebel numbers


FORMULA

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 MatulaGoebel number n can be obtained recursively in the following way: g(1)=1; if n=p(t) (=the tth prime), then g(n)=g(t)+x^G(t)*(x1)+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)).


EXAMPLE

a(5)=2 because the rooted tree with MatulaGoebel number 5 is the path tree on 4 vertices and the vertex degrees are 1,1,2,2;
a(7)=0 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y having vertices of degree 1,1,1,3.


MAPLE

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))*(x1)+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);


CROSSREFS

Cf. A182907, A190175.
Sequence in context: A160692 A051775 A108036 * A263139 A063711 A057893
Adjacent sequences: A190171 A190172 A190173 * A190175 A190176 A190177


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 09 2011


STATUS

approved



