

A190175


The GoebelMatula numbers of the rooted trees having only vertices of odd degree.


1



2, 7, 8, 28, 32, 43, 53, 98, 112, 128, 172, 212, 227, 263, 311, 343, 392, 443, 448, 512, 577, 602, 688, 742, 848, 908, 1052, 1193, 1244, 1372, 1423, 1568, 1619, 1772, 1792, 1993, 2048, 2107, 2308, 2311, 2408, 2539, 2597, 2752, 2939, 2968, 3178, 3209, 3392, 3632, 3682, 3698, 3779
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OFFSET

1,1


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.


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..53.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288
Index entries for sequences related to MatulaGoebel numbers


FORMULA

The number of vertices of even degree of the rooted trees with MatulaGoebel number n is A190174(n). The number n is in the sequence if and only if A190174(n)=0.
In A182907 one can find the generating polynomial g(n)=g(n,x) of the vertices of the rooted tree having MatulaGoebel number n, according to degree. We look for those values of n for which the polynomial g(n,x) is odd, i.e. satisfies g(n,x)=g(n,x).


EXAMPLE

7 is in the sequence because the rooted tree with MatulaGoebel number 7 is the rooted tree Y with vertices of degree 1,1,1,3.
15 is not in the sequence because the rooted tree with MatulaGoebel number 15 is the path tree ABRCDE, rooted at R; it has 2 vertices of degree 1 and 4 vertices of degree 2.


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: A := {}: for n to 4000 do if a(n) = 0 then A := `union`(A, {n}) else end if end do: A;


CROSSREFS

Cf. A182907, A190174.
Sequence in context: A117558 A117559 A236291 * A081700 A093795 A001493
Adjacent sequences: A190172 A190173 A190174 * A190176 A190177 A190178


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 30 2011, Dec 09 2011


STATUS

approved



