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A191516 Irregular triangle read by rows: a(n,k) is the number of edges with degree k (k>=1) in the rooted tree with Matula-Goebel number n (n>=3). 1
2, 2, 2, 1, 2, 1, 0, 3, 0, 3, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 0, 0, 4, 1, 2, 1, 2, 1, 2, 0, 0, 4, 1, 3, 1, 1, 3, 1, 2, 3, 2, 1, 2, 1, 0, 3, 1, 2, 4, 2, 1, 2, 3, 0, 3, 0, 4, 0, 1, 1, 3, 1, 2, 2, 2, 2, 3, 0, 0, 0, 5, 2, 4, 1, 3, 1, 1, 4, 1, 2, 0, 2, 2, 1, 0, 3, 1, 1, 0, 3, 1, 2, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 3, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

The degree of an edge is the number of edges adjacent to it.

The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel 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 Matula-Goebel numbers of the m branches of T.

Sum of entries in row n = A196050(n)  (= number of edges).

Sum(k*a(n,k), k>=1) = A198332(n)  (=sum of edge degrees (the Platt index)).

REFERENCES

A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

LINKS

Table of n, a(n) for n=3..114.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288

FORMULA

Let f(n)=F(n,x) be the generating polynomial of the edges of the rooted tree with Matula-Goebel number n, with respect to edge-degree. Write f(n)=g(n)+h(n), where g(n) is over the edges emanating from the root and h(n) is over the remaining edges. We have g(1)=0, h(1)=0; if n = p(t) (=the t-th prime), then g(n)=x^G(t), h(n)=xg(t)+h(t); if n=rs (r,s>=2), then g(n)=x^G(s)*g(r) + x^G(r)*g(s), h(n)=h(r)+h(s). G(m) denotes the number of prime divisors of m counted with multiplicities.

EXAMPLE

Row 5 is 2,1 because the rooted tree with Matula-Goebel number 5 is the path tree ABCD on 4 vertices; AB and CD have degree 1 and BC has degree 2.

Row 7 is 0,3 because the rooted tree with Matula-Goebel number 7 is Y, where no edge has degree 1 and all 3 edges have degree 2.

Triangle starts:

2;

2;

2,1;

2,1;

0,3;

0,3;

2,2;

MAPLE

with(numtheory): f := proc (n) local r, s, g, h: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: g:=proc(n) if n = 1 then 0 elif bigomega(n) = 1 then x^(bigomega(pi(n))) else x^(bigomega(s(n)))*g(r(n))+x^(bigomega(r(n)))*g(s(n)) fi end: h:=proc(n) if n=1 then 0 elif bigomega(n)=1 then x*g(pi(n))+h(pi(n)) else h(r(n))+h(s(n)) fi end: sort(expand(g(n)+h(n))) end: for n from 3 to 42 do seq(coeff(f(n), x, j), j=1..degree(f(n))) od; # yields sequence in triangular form

CROSSREFS

Cf. A196050, A198332.

Sequence in context: A320844 A212119 A096831 * A168141 A232654 A034095

Adjacent sequences:  A191513 A191514 A191515 * A191517 A191518 A191519

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Dec 15 2011

STATUS

approved

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Last modified February 23 05:58 EST 2019. Contains 320411 sequences. (Running on oeis4.)