A107373 a(n) = (n/2)*binomial(n-1, floor((n-1)/2)) - 2^(n-2).

0, 0, 1, 2, 7, 14, 38, 76, 187, 374, 874, 1748, 3958, 7916, 17548, 35096, 76627, 153254, 330818, 661636, 1415650, 2831300, 6015316, 12030632, 25413342, 50826684, 106853668, 213707336, 447472972, 894945944, 1867450648, 3734901296, 7770342787, 15540685574

Offset: 1

Author

N. J. A. Sloane, Jul 20 2007

Keywords

• nonn

Comments

Total number of descents in all faro permutations of length n-1. Faro permutations are permutations avoiding the three consecutive patterns 231, 321 and 312. They are obtained by a perfect faro shuffle of two nondecreasing words of lengths differing by at most one. See also A340567, A340568 and A340569. - Sergey Kirgizov, Jan 11 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See Table 1.
• F. Disanto and S. Rinaldi, Symmetric convex permutominoes and involutions, PU. M. A. 22:1 (2011), 39-60.
• Igor Pak, The area of cyclic polygons: Recent progress on Robbins' Conjectures, Adv. Applied Math. 34 (2005), 690-696. Special issue in memory of David Robbins.

Crossrefs

Cf. A131019, A131020, A131021.

Programs

Magma

[(n/2)*Binomial(n-1, Floor((n-1)/2)) - 2^(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 01 2013

Maple

A056040 := n -> n!/iquo(n,2)!^2:
A133265 := n -> (n+2+(n-2)*(-1)^n)/2:
A107373 := n -> (A056040(n)*A133265(n)-2^n)/4:
seq(A107373(n),n=1..34); # Peter Luschny, Aug 30 2011

Mathematica

Table[(n/2) Binomial[n-1, Floor[(n-1)/2]] - 2^(n-2), {n, 1, 40}] (* Vincenzo Librandi, Oct 01 2013 *)

Formula

a(2*n) = 2*A000531(n-1); a(2*n+1) = A000531(n). - Max Alekseyev, Sep 30 2013

(1-n)*a(n) + 2*(n-1)*a(n-1) + 4*(n-2)*a(n-2) + 8*(-n+2)*a(n-3) = 0. - R. J. Mathar, May 26 2013