%I
%S 1,1,1,2,2,2,2,2,2,2,2,2,2,3,2,2,3,2,3,3,3,3,2,3,3,3,2,3,3,3,2,3,3,3,
%T 3,2,2,3,3,3,3,2,3,4,3,3,3,3,4,3,3,2,3,4,3,3,3,3,3,3,3,3,2,4,4,2,3,4,
%U 3,3,3,3,3,4,3,4,3,3,3,4,3,3,3,4,3,4,3,3,4,3,3,4,4,3,3,4,3,4,4
%N The rounded sumconnectivity index of the rooted tree with Matula number n (n >= 2).
%C The sumconnectivity index of a graph is defined as the summation of 1/sqrt((d(u)+d(v))) over all edges uv of G, where d(w) denotes the degree of the vertex w.
%C The Matula number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the Matula 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 numbers of the m branches of T.
%D B. Zhou and N. Trinajstic, On a novel connectivity index, J. Math. Chem., 46, 2009, 12521270.
%D R. Xing, B. Zhou, and N. Trinajstic, Sumconnectivity index of molecular trees, J. Math. Chem., 48, 2010, 583591.
%D F. Goebel, On a 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
%D I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Applied Math., 160, 2012, 23142322.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%H <a href="/index/Mat#matula">Index entries for sequences related to MatulaGoebel numbers</a>
%F There are recurrence relations that give the sumconnectivity index of an "elevated" rooted tree (attach a new vertex to the root which becomes the root of the new tree) and of the merge of two rooted trees (identify the two roots). They make use of the sequence of the degrees of the level1 vertices (denoted by DL in the Maple program) .
%F In the Maple program, F(n) gives the actual (not rounded) sumconnectivity index of the rooted tree with Matula number n. For example, F(7) = 3/2; indeed, to the Matula number 7 there corresponds the star with 4 vertices, having 3 edges, each with endpoint degrees 1 and 3; then the index is 3/sqrt(4) = 3/2.
%e a(5)=2; indeed the rooted tree with Matula number 5 is the path PQRS (rooted at P). The edges PQ and RS have endpoints of degrees 1 and 2 and the edge QR has endpoints of degrees 2 and 2; consequently, the contributions of these 3 edges to the sumconnectivity index are 1/sqrt(3), 1/sqrt(3), and 1/2, respectively; the sumconnectivity index is 2/sqrt(3) + 1/2 =1.6547.
%e G.f. = x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 2*x^9 + 2*x^10 + ...
%p f := proc (x, y) options operator, arrow: 1/sqrt(x+y) end proc: c := 1/sqrt(2): with(numtheory): F := proc (n) local DL, r, s: DL := proc (n) if n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: 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 = 2 then c elif bigomega(n) = 1 then F(pi(n))(sum(f(DL(pi(n))[j], bigomega(pi(n))), j = 1 .. bigomega(pi(n))))+sum(f(DL(pi(n))[j], 1+bigomega(pi(n))), j = 1 .. bigomega(pi(n)))+f(1, 1+bigomega(pi(n))) else F(r(n))+F(s(n))(sum(f(DL(r(n))[j], bigomega(r(n))), j = 1 .. bigomega(r(n))))(sum(f(DL(s(n))[j], bigomega(s(n))), j = 1 .. bigomega(s(n))))+sum(f(DL(r(n))[j], bigomega(n)), j = 1 .. bigomega(r(n)))+sum(f(DL(s(n))[j], bigomega(n)), j = 1 .. bigomega(s(n))) end if end proc: a := proc (n) options operator, arrow: round(F(n)) end proc: seq(a(n), n = 2 .. 100);
%K nonn
%O 2,4
%A _Emeric Deutsch_, Feb 26 2014
