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%I #24 Jul 24 2023 04:53:18
%S 1,31,5119,9961471,259577085951,94554701453852671,
%T 494214691850093043122175,37747948215762478445361018961919,
%U 42694960288928350006693371507341885702143,722273364120299921501331975953872089285372151857151
%N Number of normalized graph Laplacian matrices of nonempty labeled graphs of 2n vertices that are separable in C^2 X C^n as density matrices in quantum mechanics.
%C Since separability is not invariant under graph isomorphism, all 2^(n(2n-1))-1 nonzero Laplacian matrices are treated as different. A nonzero Laplacian matrix different from the complete graph is separable in C^2 X C^n if and only if its complement is. Since the complete graph is separable, this means that a(n) is odd for all n.
%H Chai Wah Wu, <a href="http://dx.doi.org/10.1016/j.physleta.2005.10.049">Conditions for separability in generalized Laplacian matrices and diagonally dominant matrices as density matrices</a>, Physics Letters A, 351 (2006), 18-22.
%H Chai Wah Wu, <a href="http://arxiv.org/abs/1407.5663">Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics</a>, arXiv:1407.5663 [quant-ph], 2014.
%F a(n) + A245291(n) = 2^(n*(2*n-1))-1.
%F a(n) = 2^(n*(n-1))*A229865(n)-1.
%Y Cf. A245291, A229865.
%K nonn
%O 1,2
%A _Chai Wah Wu_, Jul 16 2014
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