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A307157
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a(n) is the Narumi-Katayama index of the Fibonacci cube Gamma(n).
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2
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1, 2, 24, 1152, 1399680, 290237644800, 520105859481600000000, 3435834286784202670080000000000000000, 3045775242579858715944293498880000000000000000000000000000000000
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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 Narumi-Katayama index of a connected graph is the product of the degrees of the vertices of the graph.
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LINKS
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FORMULA
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a(n) = Product_{k=1..n} 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). T(n,k) gives the number of vertices of degree k in the Fibonacci cube Gamma(n) (see A245825).
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EXAMPLE
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a(2)=2 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, the Narumi-Katayama index is 1*1*2=2.
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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(j^T(n, j), j=1..n), n=1..10);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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