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A212624 Number of vertices in all independent vertex subsets of the rooted tree with Matula-Goebel number n. 10
1, 2, 5, 5, 10, 10, 13, 13, 20, 20, 20, 23, 23, 23, 38, 33, 23, 41, 33, 45, 45, 38, 41, 55, 71, 41, 74, 48, 45, 78, 38, 81, 71, 45, 82, 92, 55, 55, 78, 105, 41, 85, 48, 82, 137, 74, 78, 131, 98, 146, 82, 85, 81, 155, 130, 108, 105, 78, 45, 173, 92, 71, 153, 193, 141, 141, 55, 98, 137, 157, 105, 212 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent.

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.

a(n) = Sum_{k>=0} k*A212623(n,k).

REFERENCES

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 Y-N. 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 Review, 10, 1968, 273.

LINKS

Table of n, a(n) for n=1..72.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

In A212623 one finds the generating polynomial P(n,x) with respect to the number of vertices of the independent vertex subsets of the rooted tree with Matula-Goebel number n. We have a(n) = subs(x=1, (d/dx)P(n,x)).

EXAMPLE

a(5)=10 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}. The total number of vertices is 10.

MAPLE

with(numtheory): A := 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 [x, 1] elif bigomega(n) = 1 then [expand(x*A(pi(n))[2]), expand(A(pi(n))[1])+A(pi(n))[2]] else [sort(expand(A(r(n))[1]*A(s(n))[1]/x)), sort(expand(A(r(n))[2]*A(s(n))[2]))] end if end proc: P := proc (n) options operator, arrow: sort(A(n)[1]+A(n)[2]) end proc: a := proc (n) options operator, arrow: subs(x = 1, diff(P(n), x)) end proc: seq(a(n), n = 1 .. 100);

CROSSREFS

Cf. A212618, A212619, A212620, A212621, A212622, A212623, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

Sequence in context: A173567 A288726 A265129 * A034387 A081240 A184443

Adjacent sequences:  A212621 A212622 A212623 * A212625 A212626 A212627

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 01 2012

STATUS

approved

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Last modified January 20 14:40 EST 2019. Contains 319333 sequences. (Running on oeis4.)