A196056 Triangle read by rows: T(n,k) is the number of nodes at level k>=1 of the rooted tree having Matula-Goebel number n (n>=2).

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 2, 2, 2, 1, 4, 1, 1, 2, 3, 2, 1, 3, 3, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 4, 1, 2, 2, 2, 2, 2, 1, 3, 3, 3, 2, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 1, 3, 1, 2, 3, 2, 3, 1, 4, 1, 1, 1, 1, 2, 1, 3, 3, 1, 2, 2, 3, 1, 1, 1, 3, 3, 1

Offset

2,4

Comments

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.

Number of entries in row n is A109082(n) (n=2,3,...).

The generating polynomial of row n is the partial Wiener polynomial of the rooted tree having Matula-Goebel number n with respect to the root (for partial Wiener polynomial see the Doslic reference, p. 68).

Links

François Marques, Table of n, a(n) for n = 2..10001 (rows 2 to 2668).

Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.

T. Doslic, Vertex-weighted Wiener polynomials for composite graphs, Ars Mathematica Contemporanea, 1, 2008, 66-80.

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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.

Index entries for sequences related to Matula-Goebel numbers

Formula

We give the recursive construction of the row generating polynomials R(n)=R(n,x). R(1)=0; if n = p(t) (=the t-th prime), then R(n)=x*R(t) + x; if n=rs (r,s>=2), then R(n)=R(r)+R(s) (2nd Maple program yields R(n)).

Example

Row n=7 is [1,2] because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 1 node at level 1 and 2 nodes at level 2.
Row n=2^m is [m] because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
Triangle starts:
1;
1,1;
2,
1,1,1;
2,1;
1,2;
3;
2,2;
2,1,1;

Maple

with(numtheory): R := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow; n/r(n) end proc: 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: for n from 2 to 45 do seq(coeff(R(n), x, k), k = 1 .. degree(R(n))) end do;
with(numtheory): R := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: 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: R(987654321);

Prog

(PARI) P(n)={ if(n==1, return(0),
    my(f=factor(n)~, v=vector(#f, k, P(primepi(f[1, k]))) );
    return( 'x*sum(k=1, #f, (v[k]+1)*f[2, k]) ) )
};
A196056_Row(n) = Vecrev(P(n))[^1]; \\ François Marques, Apr 03 2021

Crossrefs

Cf. A109082.

Sequence in context: A217467 A268057 A107435 * A161095 A276162 A118107

Adjacent sequences: A196053 A196054 A196055 * A196057 A196058 A196059

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 30 2011

Extensions

Keyword tabf added by Michel Marcus, Apr 09 2013

Status

approved