

A206493


Product, over all vertices v of the rooted tree with MatulaGoebel number n, of the number of vertices in the subtree with root v.


2



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
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OFFSET

1,2


COMMENTS

a(n) is called tree factorial. See, for example, the Brouder reference.
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


LINKS

Table of n, a(n) for n=1..65.
Ch. Brouder, RungeKutta methods and renormalization, arXiv:hepth/9904014, 1999; Eur. Phys. J. C 12, 2000, 521534.
E. 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 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Denote by V(k) the number of vertices of the rooted tree with MatulaGoebel number k. If n is the mth 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.


EXAMPLE

a(7)=12 because the rooted tree with MatulaGoebel 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 MatulaGoebel number 11 is the path tree on 5 vertices; the subtrees have 5,4,3,2,1 vertices.


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);


CROSSREFS

Sequence in context: A050125 A178667 A281881 * A304085 A302783 A008306
Adjacent sequences: A206490 A206491 A206492 * A206494 A206495 A206496


KEYWORD

nonn


AUTHOR

Emeric Deutsch, May 10 2012


STATUS

approved



