A307580 a(n) is the second multiplicative Zagreb index of the Fibonacci cube Gamma(n).

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

Offset: 1

Author

Emeric Deutsch, Apr 15 2019

Keywords

• nonn

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 the product of 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).

Examples

			a(2) = 4 because the Fibonacci cube Gamma(2) is the path-tree 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.
		

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 Institute ISSN 1840-4367, Vol. 1, 2011, 13-19.
• S. Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
• S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Crossrefs

Cf. A245825.

Programs

Maple

T := (n,k)-> add(binomial(n-2*i,k-i)*binomial(i+1,n-k-i+1), i=0..k):
seq(mul(k^(k*T(n,k)), k=1..n), n=1..7);

Formula

a(n) = Product_{k=1..n} k^(k*T(n,k)), where T(n,k) = Sum_{i=0..k} binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1).