A196059 Triangle read by rows: T(n,k) is the number of pairs of nodes at distance k in the rooted tree having Matula-Goebel number n (n>=2).

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

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 A196058(n) (n=2,3,...).

The generating polynomial of row n is the Wiener polynomial of the rooted tree having Matula-Goebel number n.

Examples

			Row n=7 is [3,3] because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having distances 1,1,1,2,2,2.
Row n=2^m is [m, m(m-1)/2] because the rooted tree with Matula-Goebel number 2^m is a star with m edges; there are m distances 1 and m(m-1)/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;
  ...
		

References

• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Links

• Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
• 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. A196056, A196058, A196060.

Programs

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);

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]]];
W[n_] := Which[n == 1, 0, PrimeOmega[n] == 1,  W[PrimePi[n]] + x*R[PrimePi[n]] + x, True, W[r[n]] + W[s[n]] + R[r[n]]*R[s[n]]];
T[n_] := Rest@CoefficientList[W[n], x];
Table[T[n], {n, 2, 35}] // Flatten (* Jean-François Alcover, Jun 19 2024, after first Maple code *)

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 = prime(t) (=the t-th prime), then W(n)=W(t)+x*R(t) + x; if n=r*s (r,s>=2), then W(n)=W(r)+W(s)+R(r)R(s) (2nd Maple program yields the Wiener polynomial W(n)).