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A184158 The sum of the odd distances in the rooted tree with Matula-Goebel number n.

%I A184158 #23 Jun 21 2024 07:34:42
%S A184158 0,1,2,2,6,6,3,3,10,10,10,10,10,10,19,4,10,17,4,14,14,19,17,14,28,17,
%T A184158 24,17,14,26,19,5,28,14,28,24,14,14,26,18,17,24,17,28,38,24,26,18,18,
%U A184158 35,28,24,5,34,44,24,18,26,14,33,24,28,31,6,40,40,14,18,38,38,18,31,24,24,52,24,37,36,28,22
%N A184158 The sum of the odd distances in the rooted tree with Matula-Goebel number n.
%C A184158 The Matula-Goebel number of a rooted tree is 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 A184158 a(n) + A184157(n) = A196051(n) (= the Wiener index of the rooted tree with Matula-Goebel number n).
%D A184158 O. Ivanciuc, T. Ivanciuc, D. J. Klein, W. A. Seitz, and A. T. Balaban, Wiener index extension by counting even/odd graph distances, J. Chem. Inf. Comput. Sci., 41, 2001, 536-549.
%H A184158 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 A184158 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 A184158 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 A184158 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 A184158 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 A184158 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A184158 a(n) is the value at x=1 of the derivative of the odd part of the Wiener polynomial W(n)=W(n,x) of the rooted tree with Matula number n. W(n) is obtained recursively in A196059. The Maple program is based on the above.
%e A184158 a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with 3 distances equal to 1.
%p A184158 with(numtheory): WP := 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(WP(pi(n))+x*R(pi(n))+x)) else sort(expand(WP(r(n))+WP(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: (1/2)*subs(x = 1, diff(WP(n), x))+(1/2)*subs(x = -1, diff(WP(n), x)) end proc: seq(a(n), n = 1 .. 80);
%t A184158 r[n_] := FactorInteger[n][[1, 1]];
%t A184158 s[n_] := n/r[n];
%t A184158 R[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*R[PrimePi[n]] + x, True,  R[r[n]] + R[s[n]]];
%t A184158 WP[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, WP[PrimePi[n]] + x*R[PrimePi[n]] + x, True, WP[r[n]] + WP[s[n]] + R[r[n]]*R[s[n]]];
%t A184158 a[n_] := (1/2)(D[WP[n], x] /. x -> 1) + (1/2)(D[WP[n], x] /. x -> -1);
%t A184158 Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Jun 21 2024, after Maple code *)
%Y A184158 Cf. A184157, A196051.
%K A184158 nonn
%O A184158 1,3
%A A184158 _Emeric Deutsch_, Oct 15 2011