

A198322


The MatulaGoebel numbers of the rooted trees that have palindromic Wiener polynomials.


0



1, 2, 7, 8, 56, 76, 107, 147, 163, 292, 454, 839, 1433, 4221, 5833, 6137, 7987, 8626, 16216, 17059, 17128, 17764, 23438, 25672, 36812, 41203, 45952, 46428, 51768, 60635, 83009, 86716, 86908, 88321, 91951, 93534, 94542, 99141, 100142, 108848, 120357, 124783, 133741, 136768, 137941, 140079, 142424, 145404, 145654
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OFFSET

1,2


COMMENTS

The Wiener polynomials are assumed to have zero constant terms.
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.
B. E. Sagan, YN. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959969.
G. Caporossi, A. A. Dobrynin, I. Gutman, and P. Hansen, Trees with palindromic Hosoya polynomials, Graph Theory Notes of New York, XXXVI, 1999, 1016.


LINKS

Table of n, a(n) for n=1..49.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

The Wiener polynomial W(n,x) of the rooted tree corresponding to the MatulaGoebel number n is given in A196059. It is palindromic if and only if x^{1+degree(W(n,x))}*W(n,1/x)=W(n,x).


EXAMPLE

7 is in the sequence because the rooted tree with MatulaGoebel number 7 is Y; 3 distances are equal to 1 and 3 distances are equal to 2; Wiener polynomial is 3x+3x^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 := {}: for n to 100000 do if expand(x^(1+degree(W(n)))*subs(x = 1/x, W(n))) = W(n) then A := `union`(A, {n}) else end if end do: A;


CROSSREFS

A196059
Sequence in context: A001493 A000637 A250715 * A252661 A222134 A011355
Adjacent sequences: A198319 A198320 A198321 * A198323 A198324 A198325


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 24 2011


STATUS

approved



