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A212624 Number of vertices in all independent vertex subsets of the rooted tree with Matula-Goebel number n.

%I A212624 #29 Jun 20 2024 11:12:16
%S A212624 1,2,5,5,10,10,13,13,20,20,20,23,23,23,38,33,23,41,33,45,45,38,41,55,
%T A212624 71,41,74,48,45,78,38,81,71,45,82,92,55,55,78,105,41,85,48,82,137,74,
%U A212624 78,131,98,146,82,85,81,155,130,108,105,78,45,173,92,71,153,193,141,141,55,98,137,157,105,212
%N A212624 Number of vertices in all independent vertex subsets of the rooted tree with Matula-Goebel number n.
%C A212624 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.
%C A212624 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 A212624 a(n) = Sum_{k>=0} k*A212623(n,k).
%H A212624 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A212624 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 A212624 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 A212624 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 A212624 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 A212624 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A212624 In A212623 one finds the generating polynomial P(n,x) with respect to the number of vertices of the independent vertex subsets of the rooted tree with Matula-Goebel number n. We have a(n) = subs(x=1, (d/dx)P(n,x)).
%e A212624 a(5)=10 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}. The total number of vertices is 10.
%p A212624 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 [x, 1] elif bigomega(n) = 1 then [expand(x*A(pi(n))[2]), expand(A(pi(n))[1])+A(pi(n))[2]] else [sort(expand(A(r(n))[1]*A(s(n))[1]/x)), sort(expand(A(r(n))[2]*A(s(n))[2]))] end if end proc: P := proc (n) options operator, arrow: sort(A(n)[1]+A(n)[2]) end proc: a := proc (n) options operator, arrow: subs(x = 1, diff(P(n), x)) end proc: seq(a(n), n = 1 .. 100);
%t A212624 r[n_] := FactorInteger[n][[1, 1]];
%t A212624 s[n_] := n/r[n];
%t A212624 A[n_] := Which[n == 1, {x, 1}, PrimeOmega[n] == 1, {x*A[PrimePi[n]][[2]], A[PrimePi[n]][[1]] + A[PrimePi[n]][[2]]}, True, {A[r[n]][[1]]*A[s[n]][[1]]/x, A[r[n]][[2]]*A[s[n]][[2]]}];
%t A212624 P[n_] := A[n] // Total;
%t A212624 a[n_] := D[P[n], x] /. x -> 1;
%t A212624 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jun 20 2024, after Maple code *)
%Y A212624 Cf. A212618, A212619, A212620, A212621, A212622, A212623, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.
%K A212624 nonn
%O A212624 1,2
%A A212624 _Emeric Deutsch_, Jun 01 2012