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%I #23 Mar 19 2024 13:43:25
%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 Connes-Moscovici weight of the rooted tree with Matula-Goebel number n. It is defined as the number of ways to build up the rooted tree from the one-vertex 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 Matula-Goebel number n by sequentially removing terminal edges.
%C 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.
%D J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, 1987, Wiley, Chichester.
%H Ch. Brouder, <a href="https://doi.org/10.1007/s100529900235">Runge-Kutta methods and renormalization</a>, Eur. Phys. J. C 12, 2000, 521-534.
%H D. J. Broadhurst and D. Kreimer, <a href="https://doi.org/10.1006/jsco.1999.0283">Renormalization automated by Hopf algebra</a>, J. Symbolic Computation, 27, 1999, 581-600.
%H Emeric Deutsch, <a href="https://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H J. Fulman, <a href="https://doi.org/10.37236/228">Mixing time for a random walk on rooted trees</a>, The Electronic J. of Combinatorics, 16, 2009, R139.
%H F. Goebel, <a href="https://doi.org/10.1016/0095-8956(80)90049-0">On a 1-1 correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H I. Gutman and A. Ivic, <a href="https://doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H I. Gutman and Y.-N. Yeh, <a href="http://elib.mi.sanu.ac.rs/files/journals/publ/73/n067p017.pdf">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H D. W. Matula, <a href="https://s2.smu.edu/~matula/Abstract-68.htm">A natural rooted tree enumeration by prime factorization</a>, SIAM Review, 10, 1968, 273.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel 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 Matula-Goebel 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 Matula-Goebel 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.
%Y A139002 is a permutation of this sequence.
%K nonn
%O 1,6
%A _Emeric Deutsch_, Jul 20 2012
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