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A307580
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a(n) is the second multiplicative Zagreb index of the Fibonacci cube Gamma(n).
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2
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1, 4, 1728, 191102976, 137105941502361600000, 27038645743755029502156994133360640000000000, 645557379413314860145212937623335060473992141864960000000000000000000000000000000000000000
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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