

A196060


The hyperWiener index of the rooted tree with MatulaGoebel number n.


3



0, 1, 5, 5, 15, 15, 12, 12, 35, 35, 35, 28, 28, 28, 70, 22, 28, 54, 22, 58, 58, 70, 54, 44, 126, 54, 90, 47, 58, 99, 70, 35, 126, 58, 108, 76, 44, 44, 99, 84, 54, 83, 47, 108, 150, 90, 99, 63, 91, 165, 108, 83, 35, 118, 210, 69, 84, 99, 58, 131, 76, 126, 129, 51, 170, 170, 44, 91, 150, 143, 84, 101, 83, 76, 231
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OFFSET

1,3


COMMENTS

The hyperWiener index of a connected graph is (1/2)*Sum [d(i,j)+d(i,j)^2], where d(i,j) is the distance between the vertices i and j and summation is over all unordered pairs of vertices (i,j).
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.


LINKS

Table of n, a(n) for n=1..75.
G. G. Cash, Relationship between the Hosoya polynomial and the hyperWiener index, Appl. Math. Letters, 15, 2002, 893895.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
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. J. Klein, I. Lukovits and I. Gutman, On the definition of the hyperWiener index for cyclecontaining structures, J. Chem. Inf. Comput. Sci., 35, 1995, 5052.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(n) = W'(n,1) + (1/2)W"(n,1), where W(n,x) is the Wiener polynomial (also called Hosoya polynomial) of the rooted tree with MatulaGoebel index n. W(n)=W(n,x) is obtained recursively in A196059. The Maple program is based on the above.


EXAMPLE

a(7)=12 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y; the distances are 1,1,1,2,2,2; sum of distances = 9; sum of squared distances = 15; (9+15)/2=12.
a(2^m) = m(3m1)/2 because the rooted tree with MatulaGoebel number 2^m is a star with m edges and we have m distances 1 and m(m1)/2 distances 2; sum of the distances = m^2; sum of the squared distances = 2m^2  m; hyperWiener index is (1/2)(3m^2  m).


MAPLE

with(numtheory): W := 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(W(pi(n))+x*R(pi(n))+x)) else sort(expand(W(r(n))+W(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: subs(x = 1, diff(W(n), x)+(1/2)*(diff(W(n), `$`(x, 2)))) end proc: seq(a(n), n = 1 .. 75);


CROSSREFS

Cf. A196059.
Sequence in context: A061200 A255304 A050350 * A147266 A147152 A189976
Adjacent sequences: A196057 A196058 A196059 * A196061 A196062 A196063


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 30 2011


STATUS

approved



