A045723 Number of configurations, excluding reflections and black-white interchanges, of n black and n white beads on a string.

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

Offset: 0

Author

Wouter Meeussen

Keywords

• nonn

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eddie Cheng and Jerrold W. Grossman, Problem 1959, Mathematics Magazine 87 (Dec. 2014), p. 396.
• L. Clarisse and P. Wocjan, On independent permutation separability criteria, Quant. Inf. Comp. 6 277-288, 2006, arXiv:quant-ph/0504160, 2005.

Crossrefs

Cf. A042942.

Cf. A027306, A042971.

Programs

Mathematica

Table[ 1/4 (2^n + Binomial[ 2 n, n ] + 2 Binomial[ -1 + n, 1/2 (-2 + n) ]*Mod[ 1 + n, 2 ]), {n, 0, 24} ]

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

Formula

a(n) = (1/4)*(2^n + C(2*n, n) + 2*C(n-1, (1/2)*(n-2))*((n+1) mod 2)).

a(n) = A042971(n) + A027306(n). - Michel Marcus, Nov 26 2018