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A184154 Triangle read by rows: T(n,k) is the number of leaves at level k>=1 of the rooted tree having Matula-Goebel number n (n>=2).

%I A184154 #28 Jun 26 2024 02:03:41
%S A184154 1,0,1,2,0,0,1,1,1,0,2,3,0,2,1,0,1,0,0,0,1,2,1,0,1,1,1,2,0,1,1,4,0,0,
%T A184154 2,1,2,0,3,2,0,1,0,3,1,0,0,1,0,0,2,3,1,0,0,2,1,1,1,0,3,2,2,0,1,0,1,1,
%U A184154 1,1,0,0,0,0,1,5,0,1,0,1,1,0,2,0,2,1,2,2,0,2,1,1,3,0,2,1,3,0,1,0,0,1,1,1,3,0,1,2,2,0,0,1,0,2,1
%N A184154 Triangle read by rows: T(n,k) is the number of leaves at level k>=1 of the rooted tree having Matula-Goebel number n (n>=2).
%C A184154 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 A184154 Number of entries in row n is A109082(n) (n=2,3,...).
%H A184154 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 A184154 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 A184154 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 A184154 D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%F A184154 We give the recursive construction of the row generating polynomials P(n)=P(n,x). P(2)=x; if n = prime(t), then P(n)=x*P(t); if n=r*s (r,s>=2), then P(n)=P(r)+P(s) (2nd Maple program yields P(n)).
%e A184154 Row n=7 is [0,2] because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 0 leaves at level 1 and 2 leaves at level 2.
%e A184154 Row n=2^m is [m] because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%e A184154 Triangle starts:
%e A184154   1;
%e A184154   0,1;
%e A184154   2;
%e A184154   0,0,1;
%e A184154   1,1;
%e A184154   0,2;
%e A184154   3;
%e A184154   0,2;
%e A184154   ...
%p A184154 with(numtheory): P := 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 = 2 then x elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: for n from 2 to 30 do seq(coeff(P(n), x, k), k = 1 .. degree(P(n))) end do; # yields sequence in triangular form
%p A184154 with(numtheory): P := 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 = 2 then x elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: for n from 2 to 30 do P(n) end do;
%t A184154 r[n_] := FactorInteger[n][[1, 1]];
%t A184154 s[n_] := n/r[n];
%t A184154 P[n_] := Which[n == 2, x, PrimeOmega[n] == 1, x*P[PrimePi[n]], True, P[r[n]] + P[s[n]]];
%t A184154 T[n_] := Rest@CoefficientList[P[n], x];
%t A184154 Table[T[n], {n, 2, 50}] // Flatten (* _Jean-François Alcover_, Jun 21 2024, after first Maple code *)
%Y A184154 Cf. A109082.
%K A184154 nonn,tabf
%O A184154 2,4
%A A184154 _Emeric Deutsch_, Oct 06 2011
%E A184154 Keyword tabf added by _Michel Marcus_, Apr 09 2013