A184167 - OEIS

%I #30 Jun 27 2024 10:20:03

%S 1,1,1,1,1,1,2,1,1,1,1,1,2,2,2,1,1,3,1,2,2,1,2,2,1,1,1,1,1,1,3,2,2,2,

%T 1,3,2,2,2,2,4,1,2,1,1,1,3,3,3,1,1,3,2,1,3,2,1,2,2,1,1,2,2,1,1,4,2,2,

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

%N Irregular triangle read by rows: T(n,k) is the number of vertices having escape distance k>=0  in the rooted tree having Matula-Goebel number n.

%C The escape distance of a vertex v in a rooted tree T is the distance from v to the nearest leaf of T that is a descendant of v. For the rooted tree ARBCDEF, rooted at R, the escape distance of B is 4 (the leaf A is not a descendant of B).

%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 Each row is nonincreasing (each vertex with escape distance k (k>=1) is the parent of some vertex with escape distance k-1).

%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F We give the recursive construction of the row generating polynomials P(n)=P(n,x): if n = prime(t), then P(n)=P(t)+x^{1+LLL(t)}; if n=r*s (r,s>=2), then P(n)=P(r)+P(s)-x^{max(LLL(r),LLL(s))}; LLL denotes the level of the lowest leaf (computed recursively and programmed in A184166) (2nd Maple program yields P(n)).

%e Row n=7 is [2,1,1] because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 2 leaves and 1 (1) vertex at distance 1 (2) from either of the leaves.

%e Triangle starts:

%e   1;

%e   1, 1;

%e   1, 1, 1;

%e   2, 1;

%e   1, 1, 1, 1;

%e   2, 2;

%e   2, 1, 1;

%e   ...

%p with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: for n to 30 do seq(coeff(P(n), x, k), k = 0 .. degree(P(n))) end do; # yields sequence in triangular form

%p with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: P(998877665544);

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

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

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

%t P[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, P[PrimePi[n]] + x^(1 + LLL[PrimePi[n]]), True, P[r[n]] + P[s[n]] - x^Max[LLL[r[n]], LLL[s[n]]]];

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

%t Table[T[n], {n, 1, 40}] // Flatten (* _Jean-François Alcover_, Jun 24 2024, after Maple code *)

%Y Cf. A184166, A184170.

%K nonn,tabf

%O 1,7

%A _Emeric Deutsch_, Oct 23 2011