%N The rounded sum-connectivity index of the rooted tree with Matula number n (n >= 2).
%C The sum-connectivity 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 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 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, 1252-1270.
%D R. Xing, B. Zhou, and N. Trinajstic, Sum-connectivity index of molecular trees, J. Math. Chem., 48, 2010, 583-591.
%D F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%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, 2314-2322.
%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 Matula-Goebel numbers</a>
%F There are recurrence relations that give the sum-connectivity 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 level-1 vertices (denoted by DL in the Maple program) .
%F In the Maple program, F(n) gives the actual (not rounded) sum-connectivity 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 sum-connectivity index are 1/sqrt(3), 1/sqrt(3), and 1/2, respectively; the sum-connectivity 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  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);
%A _Emeric Deutsch_, Feb 26 2014