

A196061


The multiplicative Wiener index of the rooted tree with MatulaGoebel number n.


0



1, 2, 2, 12, 12, 8, 8, 288, 288, 288, 144, 144, 144, 34560, 64, 144, 10368, 64, 13824, 13824, 34560, 10368, 3456, 24883200, 10368, 2985984, 5184, 13824, 4976640, 34560, 1024, 24883200, 13824, 8294400, 746496, 3456, 3456, 4976640, 1327104, 10368, 1492992, 5184, 8294400, 7166361600
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OFFSET

2,2


COMMENTS

The multiplicative Wiener index of a connected graph is the product of the distances between all unordered pairs of vertices in the graph.
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.


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, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113116.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte f. Chemie, 131, 2000, 421427.
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=2..45.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(n)=Product(k^c(k), k=1..d), where d is the diameter of the rooted tree with MatulaGoebel number n, and c(k) is the number of pairs of nodes at distance k (all these data are contained in the Wiener polynomial; see A196059). The Maple program is based on the above.


EXAMPLE

a(7)=8 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y with distances 1,1,1,2,2,2; product of distances is 8.
a(2^m) = 2^[m(m1)/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.


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: product(k^coeff(W(n), x, k), k = 1 .. degree(W(n))) end proc: seq(a(n), n = 2 .. 45);


CROSSREFS

A196059
Sequence in context: A228154 A275279 A109767 * A131121 A232853 A055772
Adjacent sequences: A196058 A196059 A196060 * A196062 A196063 A196064


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 30 2011


STATUS

approved



