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A184165 Number of independent (vertex) subsets in the rooted tree with Matula-Goebel number n.

%I A184165 #30 Jan 05 2025 19:51:39
%S A184165 2,3,5,5,8,8,9,9,13,13,13,14,14,14,21,17,14,22,17,23,23,21,22,26,34,
%T A184165 22,35,24,23,36,21,33,34,23,37,40,26,26,36,43,22,38,24,37,57,35,36,50,
%U A184165 41,59,37,38,33,62,55,44,43,36,23,66,40,34,61,65,58,58,26,41,57,62,43,76,38,40,93,44,60,60,37,83
%N A184165 Number of independent (vertex) subsets in the rooted tree with Matula-Goebel number n.
%C A184165 A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent. For example, the 1-edge tree AB has 3 independent subsets: the empty set, {A}, and {B}.
%C A184165 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 A184165 The number of independent subsets of a graph G is called the Merrifield-Simmons index of G.
%H A184165 Reinhard Zumkeller, <a href="/A184165/b184165.txt">Table of n, a(n) for n = 1..10000</a>
%H A184165 M. B. Ahmadi and M. Seif, <a href="https://www.researchgate.net/publication/266169272_The_merrifield-simmons_index_of_an_infinite_class_of_dendrimers">The Merrifield-Simmons index of an infinite class of dendrimers</a>, Digest J. of Nanomaterials and Biostructures, 5, 2010, 335-338.
%H A184165 É. Czabarka, L. Székely, and S. Wagner, <a href="http://dx.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 A184165 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 A184165 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 A184165 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 A184165 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 A184165 H. Prodinger and R. F. Tichy, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/20-1/prodinger.pdf">Fibonacci numbers of graphs</a>, Fibonacci Quarterly, 20, 1982, 16-21.
%H A184165 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A184165 Define b(n) (c(n)) to be the number of independent subsets of the rooted tree with Matula-Goebel number n that contain (do not contain) the root. We have the following recurrence for the pair A(n)=[b(n),c(n)]. A(1)=[1,1]; if n=prime(t), then A(n)=[c(t),b(t)+c(t)]; if n=r*s (r,s,>=2), then A(n)=[b(r)*b(s), c(r)*c(s)]. Clearly, a(n)=b(n)+c(n). See the Czabarka et al. reference (p. 3315, (3)). The Maple program is based on this recursive formula.
%F A184165 a(n) = A228731(n) + A228732(n). - _Reinhard Zumkeller_, Sep 01 2013
%e A184165 a(2)=3 because the tree with the Matula number 2 is the 1-edge tree AB with 3 independent subsets: (empty, {A}, {B}).
%e A184165 a(2655237841)=3216386; the tree D[3] in Fig. 1 of the Ahmadi et al. reference has Merrifield-Simmons index 3216386 (see Table 1). The Matula-Goebel number of D[3] can be found to be 227^4=2655237841.
%p A184165 with(numtheory): 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 [1, 1] elif bigomega(n) = 1 then [A(pi(n))[2], A(pi(n))[1]+A(pi(n))[2]] else [A(r(n))[1]*A(s(n))[1], A(r(n))[2]*A(s(n))[2]] end if end proc: a := proc (n) options operator, arrow: A(n)[1]+A(n)[2] end proc: seq(a(n), n = 1 .. 80);
%t A184165 r[n_] := FactorInteger[n][[1, 1]];
%t A184165 s[n_] := n/r[n];
%t A184165 A[n_] := A[n] = If[n==1, {1, 1}, If[PrimeOmega[n]==1, {A[PrimePi[n]][[2]], A[PrimePi[n]] // Total}, A[r[n]] * A[s[n]]]];
%t A184165 a[n_] := A[n] // Total;
%t A184165 a /@ Range[1, 80] (* _Jean-François Alcover_, Sep 20 2019, from Maple *)
%o A184165 (Haskell)
%o A184165 import Data.List (genericIndex)
%o A184165 a184165 n = a228731 n + a228732 n
%o A184165 a228731 n = genericIndex a228731_list (n - 1)
%o A184165 a228732 n = genericIndex a228732_list (n - 1)
%o A184165 (a228731_list, a228732_list) = unzip $ (1, 1) : map f [2..] where
%o A184165    f x | i > 0     = (a228732 i, a228731 i + a228732 i)
%o A184165        | otherwise = (a228731 u * a228731 v, a228732 u * a228732 v)
%o A184165        where i = a049084 x
%o A184165              u = a020639 x; v = x `div` u
%o A184165 -- _Reinhard Zumkeller_, Sep 01 2013
%o A184165 (PARI)
%o A184165 R(n)={my(f=factor(n), g=f); for(i=1, #f~, my([b,c]=R(primepi(f[i,1]))); f[i,1]=c; g[i,1]=b+c); [factorback(f), factorback(g)]}
%o A184165 a(n)=vecsum(R(n)); \\ _Andrew Howroyd_, Aug 01 2018
%Y A184165 Cf. A049084, A020639, A228731, A228732.
%K A184165 nonn
%O A184165 1,1
%A A184165 _Emeric Deutsch_, Oct 19 2011