A184157 The sum of the even distances in the rooted tree with Matula-Goebel number n.

0, 0, 2, 2, 4, 4, 6, 6, 10, 10, 10, 8, 8, 8, 16, 12, 8, 14, 12, 18, 18, 16, 14, 14, 28, 14, 24, 12, 18, 24, 16, 20, 28, 18, 24, 20, 14, 14, 24, 28, 14, 22, 12, 24, 34, 24, 24, 22, 30, 40, 24, 22, 20, 30, 40, 18, 28, 24, 18, 34, 20, 28, 36, 30, 36, 36, 14, 30, 34, 32, 28, 28, 22, 20, 50, 18, 42, 32, 24, 40

Offset

1,3

Comments

The Matula-Goebel number of a rooted tree is 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) + A184158(n) = A196051(n) (= the Wiener index of the rooted tree with Matula-Goebel number n).

References

O. Ivanciuc, T. Ivanciuc, D. J. Klein, W. A. Seitz, and A. T. Balaban, Wiener index extension by counting even/odd graph distances, J. Chem. Inf. Comput. Sci., 41, 2001, 536-549.

Links

Table of n, a(n) for n=1..80.

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

Formula

a(n) is the value at x=1 of the derivative of the even part of the Wiener polynomial W(n)=W(n,x) of the rooted tree with Matula number n. W(n) is obtained recursively in A196059. The Maple program is based on the above.

Example

a(7)=6 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with 3 distances equal to 2.

Maple

with(numtheory): WP := 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(WP(pi(n))+x*R(pi(n))+x)) else sort(expand(WP(r(n))+WP(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: (1/2)*subs(x = 1, diff(WP(n), x))-(1/2)*subs(x = -1, diff(WP(n), x)) end proc: seq(a(n), n = 1 .. 80);

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]]];
WP[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, WP[PrimePi[n]] + x*R[PrimePi[n]] + x, True, WP[r[n]] + WP[s[n]] + R[r[n]]*R[s[n]]];
a[n_] := (1/2)(D[WP[n], x] /. x -> 1) - (1/2)(D[WP[n], x] /. x -> -1);
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 21 2024, after Maple code *)

Crossrefs

Cf. A184158, A196051

Sequence in context: A340279 A343100 A023023 * A008643 A008644 A008645

Adjacent sequences: A184154 A184155 A184156 * A184158 A184159 A184160

Keyword

nonn

Author

Emeric Deutsch, Oct 15 2011

Status

approved