A212619 Sum of the distances between all unordered pairs of vertices of degree 3 in the rooted tree with Matula-Goebel number n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 4, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 3, 0

Offset: 1

Author

Emeric Deutsch, May 22 2012

Keywords

• nonn

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.

Examples

			a(28)=1 because the rooted tree with Matula-Goebel number 28 is obtained by joining the trees I, I, and Y at  their roots; it has 2 vertices of degree 3 (the root and the center of Y), the distance between them is 1.
a(987654321) = 22, as given by the Maple program; the reader can verify this on the rooted tree of Fig. 2 of the Deutsch reference.
		

Links

• 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.
• A. Ilic and M. Ilic, Generalizations of Wiener polarity index and terminal Wiener index, arXiv:11106.2986, 2011-2012.
• 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. A206499, A212618, A212620, A212621, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

Programs

Maple

k := 3: 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 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then sort(expand(x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) = k then sort(expand(-x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) <> k-1 and bigomega(pi(n)) <> k then sort(expand(x*g(pi(n)))) elif bigomega(s(n)) = k-1 then sort(expand(1+g(r(n))+g(s(n)))) elif bigomega(s(n)) = k then sort(expand(-1+g(r(n))+g(s(n)))) else sort(g(r(n))+g(s(n))) end if end proc; 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 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then a(pi(n))+subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) = k then a(pi(n))-subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) <> k and bigomega(pi(n)) <> k-1 then a(pi(n)) elif bigomega(s(n)) = k-1 then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))+subs(x = 1, diff(g(r(n)), x))+subs(x = 1, diff(g(s(n)), x)) elif bigomega(s(n)) = k then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))-subs(x = 1, diff(g(r(n)), x))-subs(x = 1, diff(g(s(n)), x)) else a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x)) end if end proc: seq(a(n), n = 1 .. 120);

Mathematica

k = 3;
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] := Which[n == 1, 0, PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1, Expand[x + x*g[PrimePi[n]]], PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k, Expand[-x + x*g[PrimePi[n]]], PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k - 1 && PrimeOmega[PrimePi[n]] != k,  Expand[x*g[PrimePi[n]]], PrimeOmega[s[n]] == k - 1, Expand[1 + g[r[n]] + g[s[n]]], PrimeOmega[s[n]] == k, Expand[-1 + g[r[n]] + g[s[n]]], True, g[r[n]] + g[s[n]]];
a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1, a[PrimePi[n]] + (D[g[PrimePi[n]], x] /. x -> 1), PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k, a[PrimePi[n]] - (D[g[PrimePi[n]], x] /. x -> 1), PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k && PrimeOmega[PrimePi[n]] != k - 1, a[PrimePi[n]], PrimeOmega[s[n]] == k - 1, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) + (D[g[r[n]], x] /. x -> 1) + (D[g[s[n]], x] /. x -> 1), PrimeOmega[s[n]] == k, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) - (D[g[r[n]], x] /. x -> 1) - (D[g[s[n]], x] /. x -> 1), True, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1)];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 19 2024, after Maple code *)

Formula

We give recurrence formulas for the more general case of vertices of degree k (k>=2). Let bigomega(n) denote the number of prime divisors of n, counted with multiplicities. Let g(n)=g(n,k,x) be the generating polynomial of the vertices of degree k of the rooted tree with Matula-Goebel number n with respect to level. We have a(1)=0; if n = prime(t) and bigomega(t) = k-1 then a(n) = a(t) +[dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t) =k, then a(n) = a(t) - [dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t) != k and != k-1, then a(n) = a(t); if n = r*s with r prime, s>=2, bigomega(s) =k-1, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} +[dg(r)/dx]{x=1} +[dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) =k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} - [dg(r)/dx]{x=1} - [dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) =/ k-1 and =/ k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1}.