%I
%S 1,1,1,1,1,3,1,1,3,4,1,6,3,4,10,1,1,15,1,10,10,5,3,10,10,15,15,10,4,
%T 60,1,1,15,5,20,45,6,5,45,20,3,60,4,15,105,18,10,15,10,70,15,45,1,105,
%U 35,20,15,24,1,210,15,6,105,1,105,105,1,15,63,140
%N The ConnesMoscovici weight of the rooted tree with MatulaGoebel number n. It is defined as the number of ways to build up the rooted tree from the onevertex tree by adding successively edges to the existing vertices.
%C See A206494 for the number of ways to take apart the rooted tree corresponding to the MatulaGoebel number n by sequentially removing terminal edges.
%C 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.
%D F. Goebel, On a 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
%D I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, 1987), Wiley, Chichester.
%D Ch. Brouder, RungeKutta methods and renormalization, Eur. Phys. J. C 12, 2000, 521534.
%D D. J. Broadhurst and D. Kreimer, Renormalization automated by Hopf algebra, J. Symbolic Computation, 27, 1999, 581600.
%D J. Fulman, Mixing time for a random walk on rooted trees, The Electronic J. of Combinatorics, 16, 2009, R139.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288.
%H <a href="/index/Mat#matula">Index entries for sequences related to MatulaGoebel numbers</a>
%F a(n) = V(n)!/[TF(n)*SF(n)], where V denotes "number of vertices" (A061775), TF denotes "tree factorial" (A206493), and SF denotes "symmetry factor" (A206497).
%e a(6)=3 because the rooted tree with MatulaGoebel number 6 is the path ARBC with root at R; starting with R we can obtain the tree ARBC by adding successively edges at the vertices (i) R, R, A or at (ii) R, R, B, or at (iii) R, A, R.
%e a(8)=1 because the rooted tree with MatulaGoebel number 8 is the star tree with 3 edges emanating from the root; obviously, there is only 1 way to build up this tree from the root.
%p 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: TF := 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)*TF(pi(n)) else TF(r(n))*TF(s(n))*V(n)/(V(r(n))*V(s(n))) end if end proc: SF := proc (n) if n = 1 then 1 elif nops(factorset(n)) = 1 then factorial(log[factorset(n)[1]](n))*SF(pi(factorset(n)[1]))^log[factorset(n)[1]](n) else SF(expand(op(1, ifactor(n))))*SF(expand(n/op(1, ifactor(n)))) end if end proc: a := proc (n) options operator, arrow: factorial(V(n))/(TF(n)*SF(n)) end proc: seq(a(n), n = 1 .. 120);
%Y Cf. A061775, A206493, A206497, A206494
%K nonn
%O 1,6
%A _Emeric Deutsch_, Jul 20 2012
