A206496 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.

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, 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, 35, 20, 15, 24, 1, 210, 15, 6, 105, 1, 105, 105, 1, 15, 63, 140

Offset: 1

Author

Emeric Deutsch, Jul 20 2012

Keywords

• nonn

Comments

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.

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(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.
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.
		

References

• J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, 1987, Wiley, Chichester.

Links

• Ch. Brouder, Runge-Kutta methods and renormalization, Eur. Phys. J. C 12, 2000, 521-534.
• D. J. Broadhurst and D. Kreimer, Renormalization automated by Hopf algebra, J. Symbolic Computation, 27, 1999, 581-600.
• 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 Y.-N. 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 Review, 10, 1968, 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A061775, A206493, A206497, A206494.

A139002 is a permutation of this sequence.

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: 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);

Mathematica

V[n_] := Module[{u, v}, u = FactorInteger[#][[1, 1]]&; v = #/u[#]&; If[n == 1, 1, If[PrimeQ[n], 1 + V[PrimePi[n]], V[u[n]] + V[v[n]] - 1]]];
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];
TF[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, V[n]*TF[PrimePi[n]], True, TF[r[n]]*TF[s[n]]*v[n]/(v[r[n]]*v[s[n]])];
SF[n_] := SF[n] = If[n == 1, 1, If[PrimeQ[n], SF[PrimePi[n]], Module[{p, e}, Product[{p, e} = pe; e! * SF[p]^e, {pe, FactorInteger[n]}]]]];
a[n_] := V[n]!/(TF[n]*SF[n]);
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 24 2024, after Emeric Deutsch in A061775, A206493 and A206497 *)

Formula

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).