

A307580


a(n) is the second multiplicative Zagreb index of the Fibonacci cube Gamma(n).


2



1, 4, 1728, 191102976, 137105941502361600000, 27038645743755029502156994133360640000000000, 645557379413314860145212937623335060473992141864960000000000000000000000000000000000000000
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OFFSET

1,2


COMMENTS

The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1.
The second multiplicative Zagreb index of a simple connected graph is product(deg(x))^(deg(x)) over all the vertices x of the graph (see, for example, the I. Gutman reference (p.16)).
In the Maple program, T(n,k) gives the number of vertices of degree k in the Fibonacci cube Gamma(n) (see A245825 and the KLavzar  Mollard  Petkovsek reference).


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10
I. Gutman, Multiplicative Zagreb indices of trees, Bulletin of International Mathematical Virtual Institut ISSN 18404367, Vol. 1, 2011, 1319.
S. Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505522.
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 13101322.


FORMULA

a(n) = Product_{k=1..n} k^(k*T(n,k)), where T(n,k) = Sum_{i=0..k} binomial(n2*i, ki)*binomial(i+1, nki+1).


EXAMPLE

a(2) = 4 because the Fibonacci cube Gamma(2) is the pathtree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently, a(2) = 1^1*1^1*2^2 = 4.
a(4) = 191102976 because the Fibonacci cube Gamma(4) has 5 vertices of degree 2, 2 vertices of degree 3, and 1 vertex of degree 4; consequently, a(4) = (2^2)^5*(3^3)^2*4^4 = 191102976.


MAPLE

T := (n, k)> add(binomial(n2*i, ki)*binomial(i+1, nki+1), i=0..k):
seq(mul(k^(k*T(n, k)), k=1..n), n=1..7);


CROSSREFS

Cf. A245825.
Sequence in context: A316484 A278794 A141090 * A255268 A079402 A198975
Adjacent sequences: A307577 A307578 A307579 * A307581 A307582 A307583


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Apr 15 2019


STATUS

approved



