A198325 - OEIS

%I #32 Jun 27 2024 10:19:52

%S 1,2,1,2,3,2,1,3,1,3,2,3,4,2,4,2,1,4,3,2,1,4,1,4,3,1,4,2,5,3,1,4,4,3,

%T 2,5,2,4,3,5,2,1,5,3,5,3,2,1,5,4,2,5,1,6,4,2,5,3,1,6,3,5,2,5,4,2,1,6,

%U 3,1,5,4,3,2,1,5,6,4,2,1,5,3,2,6,4,1,6,2,5,4,1,5,3,6,4,1,6,2,1,5,4,3,1,6,3,5,4,2,6,3,2,1,7,4,1

%N Irregular triangle read by rows: T(n,k) is the number of directed paths of length k (k>=1) in the rooted tree having Matula-Goebel number n (n>=2).

%C 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 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 Number of entries in row n is A109082(n) (n=2,3,...).

%C Sum of entries in row n is A196047(n).

%C Sum(k*T(n,k),k>=1)=A198326(n).

%H F. Goebel, <a href="http://dx.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="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.

%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.

%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327?seq=30">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.

%F We give the recursive construction of the row generating polynomials P(n)=P(n,x): P(1)=0; if n=prime(t), then P(n)=x*E(n)+x*P(t), where E denotes number of edges (computed recursively and programmed in A196050); if n=r*s (r,s>=2), then P(n)=P(r)+P(s) (2nd Maple program yields P(n)).

%e T(7,1)=3 and T(7,2)=2 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.

%e Triangle starts:

%e   1;

%e   2,1;

%e   2;

%e   3,2,1;

%e   3,1;

%e   3,2;

%e   3;

%e   ...

%p 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: T := proc (n, k) options operator, arrow: coeff(P(n), x, k) end proc: for n from 2 to 15 do seq(T(n, k), k = 1 .. degree(P(n))) end do; # yields sequence in triangular form

%p P(987654321); # yields P(987654321)

%t r[n_] := FactorInteger[n][[1, 1]];

%t s[n_] := n/r[n];

%t e[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, 1 + e[PrimePi[n]], True, e[r[n]] + e[s[n]]];

%t P[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*e[n] + x*P[PrimePi[n]], True, P[r[n]] + P[s[n]]];

%t T[n_] := Rest@CoefficientList[P[n], x];

%t Table[T[n], {n, 2, 50}] // Flatten (* _Jean-François Alcover_, Jun 22 2024, after Maple code *)

%Y Cf. A109082, A196047, A196050, A198326.

%K nonn,tabf

%O 2,2

%A _Emeric Deutsch_, Nov 02 2011