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A196059 Triangle read by rows: T(n,k) is the number of pairs of nodes at distance k in the rooted tree having Matula-Goebel number n (n>=2).

%I A196059 #36 Jun 25 2024 13:11:46
%S A196059 1,2,1,2,1,3,2,1,3,2,1,3,3,3,3,4,3,2,1,4,3,2,1,4,3,2,1,4,4,2,4,4,2,4,
%T A196059 4,2,5,4,3,2,1,4,6,4,4,2,5,5,4,1,4,6,5,5,3,2,5,5,3,2,5,4,3,2,1,5,5,4,
%U A196059 1,5,7,3,6,5,4,3,2,1,5,5,4,1,6,6,6,3,5,6,4,5,5,3,2,6,6,5,3,1,5,4,3,2,1,5,10,6,5,4,3,2,1,5,5,3,2,6,6,4,3,2
%N A196059 Triangle read by rows: T(n,k) is the number of pairs of nodes at distance k in the rooted tree having Matula-Goebel number n (n>=2).
%C A196059 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 A196059 Number of entries in row n is A196058(n) (n=2,3,...).
%C A196059 The generating polynomial of row n is the Wiener polynomial of the rooted tree having Matula-Goebel number n.
%D A196059 B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%H A196059 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 A196059 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 A196059 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 A196059 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 A196059 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.
%H A196059 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A196059 We give the recursive construction of the row generating polynomials W(n)=W(n,x) (the Wiener polynomials). Let R(n) be the partial Wiener polynomial with respect to the root (defined, computed and programmed in A196056). W(1)=0; if n = prime(t) (=the t-th prime), then W(n)=W(t)+x*R(t) + x; if n=r*s (r,s>=2), then W(n)=W(r)+W(s)+R(r)R(s) (2nd Maple program yields the Wiener polynomial W(n)).
%e A196059 Row n=7 is [3,3] because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having distances 1,1,1,2,2,2.
%e A196059 Row n=2^m is [m, m(m-1)/2] because the rooted tree with Matula-Goebel number 2^m is a star with m edges; there are m distances 1 and m(m-1)/2 distances 2.
%e A196059 Triangle starts:
%e A196059   1;
%e A196059   2,1;
%e A196059   2,1;
%e A196059   3,2,1;
%e A196059   3,2,1;
%e A196059   3,3;
%e A196059   3,3;
%e A196059   4,3,2,1;
%e A196059   4,3,2,1;
%e A196059   ...
%p A196059 with(numtheory): W := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc; R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc; if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(W(pi(n))+x*R(pi(n))+x)) else sort(expand(W(r(n))+W(s(n))+R(r(n))*R(s(n)))) end if end proc: for n from 2 to 35 do seq(coeff(W(n), x, k), k = 1 .. degree(W(n))) end do; # yields sequence in triangular form
%p A196059 with(numtheory): W := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc; if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(W(pi(n))+x*R(pi(n))+x)) else sort(expand(W(r(n))+W(s(n))+R(r(n))*R(s(n)))) end if end proc: W(987654321);
%t A196059 r[n_] := FactorInteger[n][[1, 1]];
%t A196059 s[n_] := n/r[n];
%t A196059 R[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*R[PrimePi[n]] + x, True,  R[r[n]] + R[s[n]]];
%t A196059 W[n_] := Which[n == 1, 0, PrimeOmega[n] == 1,  W[PrimePi[n]] + x*R[PrimePi[n]] + x, True, W[r[n]] + W[s[n]] + R[r[n]]*R[s[n]]];
%t A196059 T[n_] := Rest@CoefficientList[W[n], x];
%t A196059 Table[T[n], {n, 2, 35}] // Flatten (* _Jean-François Alcover_, Jun 19 2024, after first Maple code *)
%Y A196059 Cf. A196056, A196058, A196060.
%K A196059 nonn,tabf
%O A196059 2,2
%A A196059 _Emeric Deutsch_, Sep 30 2011