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A206499 The sum of the distances between all unordered pairs of branch vertices in the rooted tree with Matula-Goebel number n. A branch vertex is a vertex of degree >=3.

%I A206499 #32 Jun 26 2024 09:32:52
%S A206499 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
%T A206499 0,0,0,0,0,0,0,1,1,0,0,0,0,0,2,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,
%U A206499 0,1,0,0,1,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,2,1,0,0,0,0,0,4,0,0,1
%N A206499 The sum of the distances between all unordered pairs of branch vertices in the rooted tree with Matula-Goebel number n. A branch vertex is a vertex of degree >=3.
%C A206499 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 A206499 The A. Ilic and M. Ilic reference considers the statistic: the sum of the distances between all unordered pairs of vertices of degree k (see A212618, A212619).
%D A206499 F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A206499 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A206499 I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A206499 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A206499 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A206499 A. Ilic and M. Ilic, <a href="http://arxiv.org/abs/1106.2986">Generalizations of Wiener polarity index and terminal Wiener index</a>, arXiv:11106.2986.
%H A206499 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A206499 Let bigomega(n) denote the number of prime divisors of n, counted with multiplicities. Let g(n)=g(n,x) be the generating polynomial of the branch vertices of the rooted tree with Matula-Goebel number n with respect to level. We have a(1) = 0; if n = prime(t) and bigomega(t) is not 2, then a(n) = a(t); if n = prime(t) and bigomega(t) = 2, then a(n) = a(t) + [dg(t)/dx]_{x=1}; if n = r*s with r prime, bigomega(s) != 2, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1}; if n=r*s with r prime, bigomega(s)=2, then a(n)=a(r)+a(s)+ [d[g(r)g(s)]/dx]_{x=1} + [dg(r)/dx]_{x=1} + [dg(s)/dx]_{x=1}.
%e A206499 a(28)=1 because the rooted tree with Matula-Goebel number 28 is the rooted tree obtained by joining the trees I, I, and Y at their roots; it has 2 branch vertices and the distance between them is 1. a(49)=2 because the rooted tree with Matula-Goebel number 49 is the rooted tree obtained by joining two copies of Y at their roots; it has 2 branch vertices and the distance between them is 2.
%p A206499 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 0 elif bigomega(n) = 1 and bigomega(pi(n)) <> 2 then sort(expand(x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) = 2 then sort(expand(x+x*g(pi(n)))) elif bigomega(r(n))+bigomega(s(n)) = 2 then sort(expand(g(r(n))-subs(x = 0, g(r(n)))+g(s(n))-subs(x = 0, g(s(n))))) else sort(expand(g(r(n))-subs(x = 0, g(r(n)))+g(s(n))-subs(x = 0, g(s(n)))+1)) end if end proc: a := 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 elif bigomega(n) = 1 and bigomega(pi(n)) <> 2 then a(pi(n)) elif bigomega(n) = 1 and bigomega(pi(n)) = 2 then a(pi(n))+subs(x = 1, diff(g(pi(n)), x)) elif bigomega(s(n)) = 2 then a(r(n))+a(s(n))+subs(x = 1, diff((1+g(r(n)))*(1+g(s(n))), x)) else a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x)) end if end proc: seq(a(n), n = 1 .. 120);
%t A206499 r[n_] := FactorInteger[n][[1, 1]];
%t A206499 s[n_] := n/r[n];
%t A206499 g[n_] := Which[n == 1, 0, PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != 2, x*g[PrimePi[n]], PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == 2, x + x*g[PrimePi[n]], PrimeOmega[r[n]] + PrimeOmega[s[n]] == 2, g[r[n]] - (g[r[n]] /. x -> 0) + g[s[n]] - (g[s[n]] /. x -> 0), True, g[r[n]] - ( g[r[n]] /. x -> 0) + g[s[n]] - (g[s[n]] /. x -> 0) + 1];
%t A206499 a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != 2, a[PrimePi[n]], PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == 2, a[PrimePi[n]] + (D[g[PrimePi[n]], x] /. x -> 1), PrimeOmega[s[n]] == 2, a[r[n]] + a[s[n]] + (D[(1 + g[r[n]])*(1 + g[s[n]]), x] /. x -> 1), True, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1)];
%t A206499 Table[a[n], {n, 1, 101}] (* _Jean-François Alcover_, Jun 24 2024, after Maple code *)
%Y A206499 Cf. A212618, A212619.
%K A206499 nonn
%O A206499 1,49
%A A206499 _Emeric Deutsch_, May 22 2012