

A196064


The 2nd multiplicative Zagreb index of the rooted tree with MatulaGoebel number n.


1



0, 1, 4, 4, 16, 16, 27, 27, 64, 64, 64, 108, 108, 108, 256, 256, 108, 432, 256, 432, 432, 256, 432, 1024, 1024, 432, 1728, 729, 432, 1728, 256, 3125, 1024, 432, 1728, 4096, 1024, 1024, 1728, 4096, 432, 2916, 729, 1728, 6912, 1728, 1728, 12500, 2916, 6912, 1728, 2916, 3125, 16384, 4096
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OFFSET

1,3


COMMENTS

The 2nd multiplicative Zagreb index of a connected graph is the product of the products deg(i)*deg(j) taken over all edges ij of the graph (deg(v) denotes the degree of the vertex v). Alternatively, it is equal to the product of deg(i)^{deg(i)} over all vertices i of the graph.
The MatulaGoebel number of a rooted tree is 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 MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


LINKS

Table of n, a(n) for n=1..55.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman, Multiplicative Zagreb indices of trees, Bulletin of International Mathematical Virtual Institut ISSN 18404367, Vol. 1, 2011, 1319.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; a(2)=1, if n = prime(t) (the tth prime, t>=2), then a(n)=a(t)*(1+G(t))^(1+G(t))/G(t)^G(t); if n=rs (r,s>=2), then a(n)=a(r)*a(s)*G(n)^G(n)/[(G(r)^G(r))*(G(s)^G(s))]; G(m) denotes the number of prime divisors of m counted with multiplicities. The Maple program is based on this recursive formula.


EXAMPLE

a(7)=27 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y ((1*3)*(3*1)*(3*1)=27).
a(2^m) = m^m because the rooted tree with MatulaGoebel number 2^m is a star with m edges.


MAPLE

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)))^(1+bigomega(pi(n)))/bigomega(pi(n))^bigomega(pi(n)) else a(r(n))*a(s(n))*bigomega(n)^bigomega(n)/(bigomega(r(n))^bigomega(r(n))*bigomega(s(n))^bigomega(s(n))) end if end proc: seq(a(n), n = 1 .. 55);


CROSSREFS

Cf. A196065.
Sequence in context: A196065 A258722 A264039 * A220761 A278238 A218522
Adjacent sequences: A196061 A196062 A196063 * A196065 A196066 A196067


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 01 2011


STATUS

approved



