

A184157


The sum of the even distances in the rooted tree with MatulaGoebel number n.


1



0, 0, 2, 2, 4, 4, 6, 6, 10, 10, 10, 8, 8, 8, 16, 12, 8, 14, 12, 18, 18, 16, 14, 14, 28, 14, 24, 12, 18, 24, 16, 20, 28, 18, 24, 20, 14, 14, 24, 28, 14, 22, 12, 24, 34, 24, 24, 22, 30, 40, 24, 22, 20, 30, 40, 18, 28, 24, 18, 34, 20, 28, 36, 30, 36, 36, 14, 30, 34, 32, 28, 28, 22, 20, 50, 18, 42, 32, 24, 40
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OFFSET

1,3


COMMENTS

The MatulaGoebel number of a rooted tree is 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.
a(n) + A184158(n) = A196051(n) (= the Wiener index of the rooted tree with MatulaGoebel number n).


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.
O. Ivanciuc, T. Ivanciuc, D. J. Klein, W. A. Seitz, and A. T. Balaban, Wiener index extension by counting even/odd graph distances, J. Chem. Inf. Comput. Sci., 41, 2001, 536549.
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..80.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(n) is the value at x=1 of the derivative of the even part of the Wiener polynomial W(n)=W(n,x) of the rooted tree with Matula number n. W(n) is obtained recursively in A196059. The Maple program is based on the above.


EXAMPLE

a(7)=6 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y with 3 distances equal to 2.


MAPLE

with(numtheory): WP := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(WP(pi(n))+x*R(pi(n))+x)) else sort(expand(WP(r(n))+WP(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: (1/2)*subs(x = 1, diff(WP(n), x))(1/2)*subs(x = 1, diff(WP(n), x)) end proc: seq(a(n), n = 1 .. 80);


CROSSREFS

Cf. A184158, A196051
Sequence in context: A328422 A029009 A023023 * A008643 A008644 A008645
Adjacent sequences: A184154 A184155 A184156 * A184158 A184159 A184160


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 15 2011


STATUS

approved



