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

Author

Emeric Deutsch, Sep 30 2011

Keywords

• nonn
• tabf

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).

Examples

			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;
  ...
		

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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A109082.

Programs

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);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
R[n_] := Which[ n == 1, 0, PrimeOmega[n] == 1, x*R[PrimePi[n]] + x, True, R[r[n]] + R[s[n]]];
T[n_] := Rest@CoefficientList[R[n], x];
Table[T[n], {n, 2, 45}] // Flatten (* Jean-François Alcover, Jun 21 2024, after Maple code *)

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

Formula

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

Extensions

Keyword tabf added by Michel Marcus, Apr 09 2013