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A182907 Triangle read by rows: row n is the degree sequence (written in nondecreasing order) of the rooted tree with Matula-Goebel number n.

%I A182907 #28 Jun 25 2024 17:06:42
%S A182907 0,1,1,1,1,2,1,1,2,1,1,2,2,1,1,2,2,1,1,1,3,1,1,1,3,1,1,2,2,2,1,1,2,2,
%T A182907 2,1,1,2,2,2,1,1,1,2,3,1,1,1,2,3,1,1,1,2,3,1,1,2,2,2,2,1,1,1,1,4,1,1,
%U A182907 1,2,3,1,1,1,2,2,3,1,1,1,1,4,1,1,1,2,2,3,1,1,1,2,2,3,1,1,2,2,2,2,1,1,1,2,2,3,1,1,1,1,2,4,1,1,2,2,2,2,2
%N A182907 Triangle read by rows: row n is the degree sequence (written in nondecreasing order) of the rooted tree with Matula-Goebel number n.
%C A182907 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 A182907 The number of entries in row n is A061775(n) (= number of vertices of the rooted tree).
%H A182907 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 A182907 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 A182907 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 A182907 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 A182907 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 A182907 For a graph with degree sequence a,b,c,..., define the degree sequence polynomial to be x^a + x^b + x^c + ... . The degree sequence polynomial g(n)=g(n,x) of the rooted tree with Matula-Goebel number n can be obtained recursively in the following way: g(1)=1; if n=prime(t) (=the t-th prime), then g(n)=g(t)+x^G(t)*(x-1)+x; if n=r*s (r,s>=2), then g(n)=g(r)+g(s)-x^G(r)-x^G(s)+x^G(n); G(m) is the number of prime divisors of m counted with multiplicities. The Maple program, based on this recursive procedure, finds for an arbitrary n the polynomial g(n,x) and then extracts from this polynomial the degree sequence S(n).
%e A182907 Row 7 is 1,1,1,3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
%e A182907 Row 32 is 1,1,1,1,1,5 because the rooted tree with Matula-Goebel number 32 is a star with 5 edges.
%p A182907 with(numtheory): g := 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 sort(expand(g(pi(n))+x^bigomega(pi(n))*(x-1)+x)) else sort(expand(g(r(n))+g(s(n))-x^bigomega(r(n))-x^bigomega(s(n))+x^bigomega(n))) end if end proc: S := proc (n) if n = 1 then 0 else seq(seq(j, i = 1 .. coeff(g(n), x, j)), j = 1 .. degree(g(n))) end if end proc: for n to 25 do S(n) end do; # yields sequence in triangular form
%t A182907 r[n_] := FactorInteger[n][[1, 1]];
%t A182907 s[n_] := n/r[n];
%t A182907 g[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, g[PrimePi[n]] + x^PrimeOmega[PrimePi[n]]*(x - 1) + x, True, g[r[n]] + g[s[n]] - x^PrimeOmega[r[n]] - x^PrimeOmega[s[n]] + x^PrimeOmega[n]];
%t A182907 S[n_] := If[n == 1, {0}, Table[t = Table[j, {i, 1, Coefficient[g[n], x, j]}]; If[t == {}, Nothing, t], {j, 1, Exponent[g[n], x]}]] // Flatten;
%t A182907 Table[S[n], {n, 1, 25}] // Flatten (* _Jean-François Alcover_, Jun 20 2024, after Maple code *)
%Y A182907 Cf. A061775.
%K A182907 nonn,tabf
%O A182907 1,6
%A A182907 _Emeric Deutsch_, Oct 05 2011