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A045723
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Number of configurations, excluding reflections and black-white interchanges, of n black and n white beads on a string.
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7
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1, 1, 3, 7, 23, 71, 252, 890, 3299, 12283, 46508, 176870, 677294, 2602198, 10034104, 38787572, 150289699, 583434323, 2268861516, 8836447022, 34461940538, 134564992898, 526025965864, 2058359779052, 8061905791118, 31602659998046
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OFFSET
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0,3
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COMMENTS
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This sequence (with offset 0) equals the probable number of inequivalent classes of permutations acting on an n-party state under the trace norm in the context of permutation criteria for separability. - Lieven Clarisse, Apr 28 2006
Number of connected components of an undirected graph where the nodes are the n-subsets of {1,...,2n} and an edge (A,B) appears if B = {1,...,2n} \ A or B = {2n + 1 - i: i in A}. See Mathematics Magazine link. - Rob Pratt, Aug 10 2015
Number of distinct staircase walks connecting opposite corners of a square grid of side n > 1. - Christian Barrientos, Nov 25 2018
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LINKS
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Eddie Cheng and Jerrold W. Grossman, Problem 1959, Mathematics Magazine 87 (Dec. 2014), p. 396.
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FORMULA
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a(n) = (1/4)*(2^n + C(2*n, n) + 2*C(n-1, (1/2)*(n-2))*((n+1) mod 2)).
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MATHEMATICA
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Table[ 1/4 (2^n + Binomial[ 2 n, n ] + 2 Binomial[ -1 + n, 1/2 (-2 + n) ]*Mod[ 1 + n, 2 ]), {n, 0, 24} ]
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PROG
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(PARI) a(n) = (1/4)*(2^n + binomial(2*n, n) + if ((n+1)%2, 2*binomial(n-1, (1/2)*(n-2)))); \\ Michel Marcus, Nov 25 2018
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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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