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A193403 Triangle read by rows: row n contains the coefficients (of the increasing powers of the variable x) of the matching polynomial of the rooted tree having Matula-Goebel number n.

%I A193403 #47 Jun 26 2025 07:30:30
%S A193403 0,1,-1,0,1,0,-2,0,1,0,-2,0,1,1,0,-3,0,1,1,0,-3,0,1,0,0,-3,0,1,0,0,-3,
%T A193403 0,1,0,3,0,-4,0,1,0,3,0,-4,0,1,0,3,0,-4,0,1,0,2,0,-4,0,1,0,2,0,-4,0,1,
%U A193403 0,2,0,-4,0,1,-1,0,6,0,-5,0,1,0,0,0,-4,0,1,0,2,0,-4,0,1,-1,0,5,0,-5,0,1,0,0,0,-4,0,1
%N A193403 Triangle read by rows: row n contains the coefficients (of the increasing powers of the variable x) of the matching polynomial of the rooted tree having Matula-Goebel number n.
%C A193403 Row n contains 1+A061775(n) entries (= 1 + number of vertices of the rooted tree).
%C A193403 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 A193403 After activating the Maple program, the command mm(n) will yield the matching polynomial of the rooted tree corresponding to the Matula-Goebel number n.
%D A193403 C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
%H A193403 É. Czabarka, L. Székely, and S. Wagner, <a href="https://doi.org/10.1016/j.dam.2009.07.004">The inverse problem for certain tree parameters</a>, Discrete Appl. Math., 157, 2009, 3314-3319.
%H A193403 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A193403 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 A193403 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 A193403 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 A193403 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 A193403 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A193403 Define b(n) (c(n)) to be the generating polynomials of the matchings of the rooted tree with Matula-Goebel number n that contain (do not contain) the root, with respect to the size of the matching. We have the following recurrence for the pair M(n)=[b(n),c(n)]. M(1)=[0,1]; if n=prime(t) (=the t-th prime), then M(n)=[xc(t),b(t)+c(t)]; if n=r*s (r,s,>=2), then M(n)=[b(r)*c(s)+c(r)*b(s), c(r)*c(s)]. Then m(n)=b(n)+c(n) is the generating polynomial of the matchings of the rooted tree with respect to the size of the matchings (a modified matching polynomial). The actual matching polynomial is obtained by the substitution x = -1/x^2, followed by multiplication by x^N(n), where N(n) is the number of vertices of the rooted tree.
%p A193403 with(numtheory): N := 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 = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: M := 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 = 1 then [0, 1] elif bigomega(n) = 1 then [x*M(pi(n))[2], M(pi(n))[1]+M(pi(n))[2]] else [M(r(n))[1]*M(s(n))[2]+M(r(n))[2]*M(s(n))[1], M(r(n))[2]*M(s(n))[2]] end if end proc: m := proc (n) options operator, arrow: sort(expand(M(n)[1]+M(n)[2])) end proc: mm := proc (n) options operator, arrow: sort(expand(x^N(n)*subs(x = -1/x^2, m(n)))) end proc: for n to 19 do seq(coeff(mm(n), x, j), j = 0 .. N(n)) end do; # yields sequence in triangular form
%t A193403 r[n_] := FactorInteger[n][[1, 1]];
%t A193403 s[n_] := n/r[n];
%t A193403 V[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + V[PrimePi[n]], True,  V[r[n]] + V[s[n]] - 1];
%t A193403 M[n_] := Which[n == 1, {0, 1}, PrimeOmega[n] == 1, {x*M[PrimePi[n]][[2]], M[PrimePi[n]][[1]] + M[PrimePi[n]][[2]]}, True, {M[r[n]][[1]]*M[s[n]][[2]] + M[r[n]][[2]]*M[s[n]][[1]], M[r[n]][[2]]*M[s[n]][[2]]}];
%t A193403 m[n_] := M[n] // Total;
%t A193403 mm[n_] := x^V[n]*(m[n] /. x -> -1/x^2);
%t A193403 T[n_] := CoefficientList[mm[n], x];
%t A193403 Table[T[n], {n, 1, 19}] // Flatten (* _Jean-François Alcover_, Jun 21 2024, after Maple code *)
%o A193403 (Sage)
%o A193403 def M(n) :
%o A193403     if n == 1 : return [0, 1, 1]
%o A193403     if 1 == sloane.A001222(n) : # bigomega
%o A193403         mpi = M(prime_pi(n))
%o A193403         return [x*mpi[1], mpi[0]+mpi[1], 1+mpi[2]]
%o A193403     r = max(prime_divisors(n)); mr = M(r); ms = M(n//r)
%o A193403     return [mr[0]*ms[1]+mr[1]*ms[0],mr[1]*ms[1],mr[2]+ms[2]-1]
%o A193403 def A193403_coeffs(n) :
%o A193403     mn = M(n)
%o A193403     q = (mn[0]+mn[1]).subs(x=-1/x^2)
%o A193403     p = expand(x^mn[2]*q)
%o A193403     return coefficient_list(p, x)
%o A193403 for n in (1..19) : print(A193403_coeffs(n))  # _Peter Luschny_, Feb 12 2012
%Y A193403 Cf. A061775, A202853.
%K A193403 sign,tabf
%O A193403 1,7
%A A193403 _Emeric Deutsch_, Feb 12 2012