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

%I A307580 #17 Oct 17 2024 04:52:28
%S A307580 1,4,1728,191102976,137105941502361600000,
%T A307580 27038645743755029502156994133360640000000000,
%U A307580 645557379413314860145212937623335060473992141864960000000000000000000000000000000000000000
%N A307580 a(n) is the second multiplicative Zagreb index of the Fibonacci cube Gamma(n).
%C A307580 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.
%C A307580 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).
%C A307580 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).
%H A307580 Alois P. Heinz, <a href="/A307580/b307580.txt">Table of n, a(n) for n = 1..10</a>
%H A307580 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 Institute ISSN 1840-4367, Vol. 1, 2011, 13-19.
%H A307580 S. Klavžar, <a href="http://dx.doi.org/10.1007/s10878-011-9433-z">Structure of Fibonacci cubes: a survey</a>, J. Comb. Optim., 25, 2013, 505-522.
%H A307580 S. Klavžar, M. Mollard and M. Petkovšek, <a href="https://doi.org/10.1016/j.disc.2011.03.019">The degree sequence of Fibonacci and Lucas cubes</a>, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.
%F A307580 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).
%e A307580 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.
%e A307580 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.
%p A307580 T := (n,k)-> add(binomial(n-2*i,k-i)*binomial(i+1,n-k-i+1), i=0..k):
%p A307580 seq(mul(k^(k*T(n,k)), k=1..n), n=1..7);
%Y A307580 Cf. A245825.
%K A307580 nonn
%O A307580 1,2
%A A307580 _Emeric Deutsch_, Apr 15 2019