A206493 Product, over all vertices v of the rooted tree with Matula-Goebel number n, of the number of vertices in the subtree with root v.

1, 2, 6, 3, 24, 8, 12, 4, 20, 30, 120, 10, 40, 15, 72, 5, 60, 24, 20, 36, 36, 144, 120, 12, 252, 48, 56, 18, 180, 84, 720, 6, 336, 72, 126, 28, 60, 24, 112, 42, 240, 42, 90, 168, 192, 140, 504, 14, 63, 288, 168, 56, 30, 64, 1152, 21, 56, 210, 360, 96, 168, 840, 96, 7, 384

Offset: 1

Author

Emeric Deutsch, May 10 2012

Keywords

• nonn

Comments

a(n) is called tree factorial. See, for example, the Brouder reference.

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(7)=12 because the rooted tree with Matula-Goebel number 7 is Y; denoting the vertices in preorder by a,b,c, and d, the number of vertices of the subtrees having these roots are 4, 3, 1, and 1, respectively. a(11)=120 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices; the subtrees have 5,4,3,2,1 vertices.
		

Links

• Ch. Brouder, Runge-Kutta methods and renormalization, arXiv:hep-th/9904014, 1999; Eur. Phys. J. C 12, 2000, 521-534.
• Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
• J. Fulman, Mixing time for a random walk on rooted trees, The Electronic J. of Combinatorics, 16, 2009, R139.
• 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.
• Kevin Ryde, PARI/GP Code
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A196068 (sum of subtree sizes).

Programs

Maple

with(numtheory): V := 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 1+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: H := 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 V(n)*H(pi(n)) else H(r(n))*H(s(n))*V(n)/(V(r(n))*V(s(n))) end if end proc: seq(H(n), n = 1 .. 100);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
V[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + V[PrimePi[n]], True, V[r[n]] + V[s[n]] - 1];
H[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, V[n]*H[PrimePi[n]], True,  H[r[n]]*H[s[n]]*V[n]/(V[r[n]]*V[s[n]])];
Table[H[n], {n, 1, 100}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)

PARI

\\ See links.

Formula

Denote by V(k) the number of vertices of the rooted tree with Matula-Goebel number k. If n is the m-th prime, then a(n) = a(m)*V(n); if n=rs, r,s>=2, then a(n) = a(r)a(s)V(n)/{V(r)V(s)}. The Maple program is based on these recurrence relations.