

A196066


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


0



0, 0, 2, 2, 8, 8, 3, 3, 20, 20, 20, 12, 12, 12, 40, 4, 12, 29, 4, 28, 28, 40, 29, 17, 70, 29, 36, 16, 28, 55, 40, 5, 70, 28, 53, 40, 17, 17, 55, 38, 29, 38, 16, 53, 68, 36, 55, 23, 36, 93, 53, 38, 5, 48, 112, 21, 38, 55, 28, 73, 40, 70, 45, 6, 92, 92, 17, 36, 68, 70, 38, 53, 38, 40, 114, 21, 89, 72, 53, 50
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OFFSET

1,3


COMMENTS

The reverse Wiener index of a connected graph is (1/2)N(N1)D  W, where N, D, and W are, respectively, the number of vertices, the diameter, and the Wiener index of 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 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.
A. T. Balaban, D. Mills, O. Ivanciuc, and S. C. Basak, Reverse Wiener indices, Croatica Chemica Acta, 73 (4), 2000, 923941.


LINKS

Table of n, a(n) for n=1..80.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(n)=(1/2)N(n)*(N(n)1)*d(n)  W(n), where N, d, and W are, respectively, the number of vertices, the diameter, and the Wiener index of the rooted tree with MatulaGoebel number n (all these data are contained in the Wiener polynomial; see A196059). The Maple program is based on the above.


EXAMPLE

a(7)=3 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y with N=4, d=2, W=9 (distances are 1,1,1,2,2,2); (1/2)*4*3*29 = 3.


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: N := proc (n) options operator, arrow: 1+coeff(Wp(n), x) end proc: d := proc (n) options operator, arrow: degree(Wp(n)) end proc: W := proc (n) options operator, arrow: subs(x = 1, diff(Wp(n), x)) end proc: a := proc (n) options operator, arrow: (1/2)*N(n)*(N(n)1)*d(n)W(n) end proc: 0, seq(a(n), n = 2 .. 80);


CROSSREFS

Cf. A196059.
Sequence in context: A195138 A094887 A021441 * A260825 A138102 A187791
Adjacent sequences: A196063 A196064 A196065 * A196067 A196068 A196069


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 01 2011


STATUS

approved



