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A198326 Sum of lengths of all directed paths in the rooted tree having Matula-Goebel number n.

%I A198326 #14 Jun 22 2024 14:11:27
%S A198326 0,1,4,2,10,5,7,3,8,11,20,6,13,8,14,4,16,9,10,12,11,21,19,7,20,14,12,
%T A198326 9,23,15,35,5,24,17,17,10,16,11,17,13,26,12,19,22,18,20,29,8,14,21,20,
%U A198326 15,13,13,30,10,14,24,30,16,22,36,15,6,23,25,22,18,23,18,26,11,25,17,24,12,27,18,38,14,16,27,36,13,26,20,27,23,19,19
%N A198326 Sum of lengths of all directed paths in the rooted tree having Matula-Goebel number n.
%C A198326 A directed path of length k in a rooted tree is a sequence of k+1 vertices v[1], v[2], ..., v[k], v[k+1], such that v[j] is a child of v[j-1] for j = 2,3,...,k+1.
%C A198326 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.
%C A198326 a(n)=Sum(k*A198325(n,k), k>=1).
%D A198326 F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A198326 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A198326 I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A198326 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A198326 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A198326 In A198325 we give the recursive construction of the generating polynomials P(n)=P(n,x) of the directed paths of the rooted tree corresponding to the Matula-Goebel number n, with respect to length. a(n) is the derivative dP(n,x)/dx, evaluated at x=1.
%e A198326 a(7)=7 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 3 directed paths of length 1 (the edges) and 2 directed paths of length 2.
%p A198326 with(numtheory): P := proc (n) local r, s, E: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*E(n)+x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: seq(subs(x = 1, diff(P(n), x)), n = 1 .. 90);
%t A198326 r[n_] := FactorInteger[n][[1, 1]];
%t A198326 s[n_] := n/r[n];
%t A198326 e[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, 1 + e[PrimePi[n]], True,  e[r[n]] + e[s[n]]];
%t A198326 P[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*e[n] + x*P[PrimePi[n]], True, P[r[n]] + P[s[n]]];
%t A198326 a[n_] := D[P[n], x] /. x -> 1;
%t A198326 Table[a[n], {n, 1, 90}] (* _Jean-François Alcover_, Jun 22 2024, after Maple code *)
%Y A198326 Cf. A198325.
%K A198326 nonn
%O A198326 1,3
%A A198326 _Emeric Deutsch_, Nov 02 2011