

A198325


Irregular triangle read by rows: T(n,k) is the number of directed paths of length k (k>=1) in the rooted tree having MatulaGoebel number n (n>=2).


1



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

2,2


COMMENTS

A directed path of length k in a rooted tree is a sequence of k+1 vertices v[1], v[2], ..., v[k], v[k+1], such that v[j] is a child of v[j1] for j = 2,3,...,k+1.
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 A109082(n) (n=2,3,...).
Sum of entries in row n is A196047(n).
Sum(k*T(n,k),k>=1)=A198326(n).


REFERENCES

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..116.
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.


FORMULA

We give the recursive construction of the row generating polynomials P(n)=P(n,x): P(1)=0; if n=p(t) (=the tth prime), then P(n)=x*E(n)+x*P(t), where E denotes number of edges (computed recursively and programmed in A196050); if n=rs (r,s>=2), then P(n)=P(r)+P(s) (2nd Maple program yields P(n)).


EXAMPLE

T(7,1)=3 and T(7,2)=2 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y, having 3 directed paths of length 1 (the edges) and 2 directed paths of length 2.
Triangle starts:
1;
2,1;
2;
3,2,1;
3,1;
3,2;
3;


MAPLE

with(numtheory): P := proc (n) local r, s, E: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*E(n)+x*P(pi(n)))) else sort(P(r(n)) +P(s(n))) end if end proc: T := proc (n, k) options operator, arrow: coeff(P(n), x, k) end proc: for n from 2 to 15 do seq(T(n, k), k = 1 .. degree(P(n))) end do; # yields sequence in triangular form
P(987654321); # yields P(987654321)


CROSSREFS

Cf. A109082, A196047, A196050, A198326.
Sequence in context: A182110 A175328 A338776 * A293909 A002850 A111944
Adjacent sequences: A198322 A198323 A198324 * A198326 A198327 A198328


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Nov 02 2011


STATUS

approved



