

A196059


Triangle read by rows: T(n,k) is the number of pairs of nodes at distance k in the rooted tree having MatulaGoebel number n (n>=2).


8



1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 3, 3, 3, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 4, 2, 4, 4, 2, 4, 4, 2, 5, 4, 3, 2, 1, 4, 6, 4, 4, 2, 5, 5, 4, 1, 4, 6, 5, 5, 3, 2, 5, 5, 3, 2, 5, 4, 3, 2, 1, 5, 5, 4, 1, 5, 7, 3, 6, 5, 4, 3, 2, 1, 5, 5, 4, 1, 6, 6, 6, 3, 5, 6, 4, 5, 5, 3, 2, 6, 6, 5, 3, 1, 5, 4, 3, 2, 1, 5, 10, 6, 5, 4, 3, 2, 1, 5, 5, 3, 2, 6, 6, 4, 3, 2
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OFFSET

2,2


COMMENTS

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.
Number of entries in row n is A196058(n) (n=2,3,...).
The generating polynomial of row n is the Wiener polynomial of the rooted tree having 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.
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.


LINKS

Table of n, a(n) for n=2..121.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to MatulaGoebel numbers


FORMULA

We give the recursive construction of the row generating polynomials W(n)=W(n,x) (the Wiener polynomials). Let R(n) be the partial Wiener polynomial with respect to the root (defined, computed and programmed in A196056). W(1)=0; if n = p(t) (=the tth prime), then W(n)=W(t)+x*R(t) + x; if n=rs (r,s>=2), then W(n)=W(r)+W(s)+R(r)R(s) (2nd Maple program yields the Wiener polynomial W(n)).


EXAMPLE

Row n=7 is [3,3] because the rooted tree with MatulaGoebel number 7 is the rooted tree Y, having distances 1,1,1,2,2,2.
Row n=2^m is [m, m(m1)/2] because the rooted tree with MatulaGoebel number 2^m is a star with m edges; there are m distances 1 and m(m1)/2 distances 2.
Triangle starts:
1;
2,1;
2,1;
3,2,1;
3,2,1;
3,3;
3,3;
4,3,2,1;
4,3,2,1;


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: for n from 2 to 35 do seq(coeff(W(n), x, k), k = 1 .. degree(W(n))) end do; # yields sequence in triangular form
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: W(987654321);


CROSSREFS

Cf. A196056, A196058, A196060.
Sequence in context: A232502 A288738 A214651 * A272900 A023116 A084822
Adjacent sequences: A196056 A196057 A196058 * A196060 A196061 A196062


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Sep 30 2011


STATUS

approved



