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%I #17 Apr 16 2019 02:50:39
%S 0,1,4,4,16,16,9,9,64,64,64,36,36,36,256,16,36,144,16,144,144,256,144,
%T 64,1024,144,576,81,144,576,256,25,1024,144,576,256,64,64,576,256,144,
%U 324,81,576,2304,576,576,100,324,2304,576,324,25,1024,4096,144,256,576,144,1024,256,1024,1296,36,2304
%N The 1st multiplicative Zagreb index of the rooted tree with Matula-Goebel number n.
%C The 1st multiplicative Zagreb index of a connected graph is the product of the squared degrees of the vertices of the graph. Alternatively, it is the square of the Narumi-Katayama index.
%C 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.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%H 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 I. Gutman, <a href="http://www.imvibl.org/buletin/bulletin_1_13_19.pdf">Multiplicative Zagreb indices of trees</a>, Bulletin of International Mathematical Virtual Institut ISSN 1840-4367, Vol. 1, 2011, 13-19.
%H 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 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 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 H. Narumi and M. Katayama, <a href="http://hdl.handle.net/2115/38010">Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons</a>, Mem. Fac. Engin. Hokkaido Univ., 16, 1984, 209-214.
%H Z. Tomovic and I. Gutman, <a href="https://www.shd.org.rs/JSCS/Vol66/No4.htm#Narumi">Narumi-Katayama index of phenylenes</a>, J. Serb. Chem. Soc., 66(4), 2001, 243-247.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(1)=0; a(2)=1, if n = prime(t) (the t-th prime, t>=2), then a(n)=a(t)*(1+G(t))^2/G(t)^2; if n=rs (r,s>=2), then a(n)=a(r)*a(s)*G(n)^2/[G(r)*G(s)]^2; G(m) denotes the number of prime divisors of m counted with multiplicities. The Maple program is based on this recursive formula.
%F a(n) = (A196063(n))^2.
%e a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*9*1*1=9).
%e a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%p 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 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))*(1+bigomega(pi(n)))^2/bigomega(pi(n))^2 else a(r(n))*a(s(n))*bigomega(n)^2/(bigomega(r(n))^2*bigomega(s(n))^2) end if end proc: seq(a(n), n = 1 .. 65);
%Y Cf. A196063, A196064.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Oct 01 2011
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