A190174 Number of vertices of even degree in the rooted tree with Matula-Goebel number n.

1, 0, 1, 1, 2, 2, 0, 0, 3, 3, 3, 1, 1, 1, 4, 1, 1, 2, 1, 2, 2, 4, 2, 2, 5, 2, 3, 0, 2, 3, 4, 0, 5, 2, 3, 3, 2, 2, 3, 3, 2, 1, 0, 3, 4, 3, 3, 1, 1, 4, 3, 1, 0, 4, 6, 1, 3, 3, 2, 4, 3, 5, 2, 1, 4, 4, 2, 1, 4, 2, 3, 2, 1, 3, 5, 1, 4, 2, 3, 2, 5, 3, 3, 2, 4, 1

Offset

1,5

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.

The degree sequences of the rooted trees with Matula-Goebel number n are given in A182907.

Links

Table of n, a(n) for n=1..86.

Emeric Deutsch, Rooted tree statistics from Matula numbers, 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

Formula

For a graph with degree sequence a,b,c,..., define the degree sequence polynomial to be x^a + x^b + x^c + ... . The degree sequence polynomial g(n)=g(n,x) of the rooted tree with Matula-Goebel number n can be obtained recursively in the following way: g(1)=1; if n=prime(t), then g(n)=g(t)+x^G(t)*(x-1)+x; if n=r*s (r,s>=2), then g(n)=g(r)+g(s)-x^G(r)-x^G(s)+x^G(n); G(m) is the number of prime divisors of m counted with multiplicities. Clearly, a(n)=(1/2)*(g(n,1) + g(n,-1)).

Example

a(5)=2 because the rooted tree with Matula-Goebel number 5 is the path tree on 4 vertices and the vertex degrees are 1,1,2,2;
a(7)=0 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y having vertices of degree 1,1,1,3.

Maple

with(numtheory): g := 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 1 elif bigomega(n) = 1 then sort(expand(g(pi(n))+x^bigomega(pi(n))*(x-1)+x)) else sort(expand(g(r(n))+g(s(n))-x^bigomega(r(n))-x^bigomega(s(n))+x^bigomega(n))) end if end proc: a := proc (n) options operator, arrow: (1/2)*subs(x = 1, g(n))+(1/2)*subs(x = -1, g(n)) end proc: seq(a(n), n = 1 .. 110);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] = Which[n == 1, 1, PrimeOmega[n] == 1, g[PrimePi[n]] + x^PrimeOmega[PrimePi[n]]*(x - 1) + x , True, g[r[n]] + g[s[n]] - x^PrimeOmega[r[n]] - x^PrimeOmega[s[n]] + x^PrimeOmega[n]];
a[n_] := (1/2)(g[n] /. x -> 1) + (1/2)(g[n] /. x -> -1);
Table[a[n], {n, 1, 110}] (* Jean-François Alcover, Jun 20 2024, after Maple code *)

Crossrefs

Cf. A182907, A190175.

Sequence in context: A160692 A051775 A108036 * A263139 A063711 A057893

Adjacent sequences: A190171 A190172 A190173 * A190175 A190176 A190177

Keyword

nonn

Author

Emeric Deutsch, Dec 09 2011

Status

approved