A285850 Number of ways n couples can sit in a row such that exactly one couple sits next to each other.

0, 2, 8, 288, 15744, 1401600, 183582720, 33223034880, 7939197665280, 2421184409763840, 917547530747904000, 422959572499916390400, 233037523912020826521600, 151234400024881955183001600, 114177664785555609793383628800, 99217287255932372662490234880000

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..224

T. Amdeberhan et al., n-distant permutations more than not, MathOverflow, 2017.

Formula

For n>0, a(n) = A007060(n) + 2*n*A007060(n-1).

For n>1, a(n) = ( (4*n^2 - 8*n + 1)*a(n-1) + (2*n-2)*(2*n-1)*a(n-2) ) * 2*n/(2*n-3).

(12*n^3+84*n^2+192*n+144)*a(n+1)+(8*n^3+34*n^2-6*n-108)*a(n+2)+(-4*n^3-42*n^2-147*n-162)*a(n+3)+(n+3)*a(n+4) = 0. - Robert Israel, Apr 28 2017

Example

For n=2, if the two couples are (1,2) and (a,b), the a(2) = 8 solutions are a12b, a21b, b12a, b21a, 1ab2, 1ba2, 2ab1, 2ba1. - N. J. A. Sloane, Apr 28 2017

Maple

f:= rectoproc({(12*x^3+84*x^2+192*x+144)*a(x+1)+(8*x^3+34*x^2-6*x-108)*a(x+2)+(-4*x^3-42*x^2-147*x-162)*a(x+3)+(x+3)*a(x+4), a(0) = 0, a(1) = 2, a(2) = 8, a(3) = 288}, a(x), remember):
map(f, [$0..50]); # Robert Israel, Apr 28 2017

Mathematica

a007060[n_]:=Sum[(-1)^(n - k) Binomial[n, k] Subfactorial[2k], {k, 0, n}]; a[n_]:=If[n<1, 0, a007060[n] + 2n*a007060[n - 1]]; Table[a[n], {n, 0, 50}] (* Indranil Ghosh, Apr 28 2017 *)

Prog

(Python)
from sympy import binomial, subfactorial
def a007060(n): return sum([(-1)**(n - k)*binomial(n, k)*subfactorial(2*k) for k in range(n + 1)])
def a(n): return 0 if n<1 else a007060(n) + 2*n*a007060(n - 1) # Indranil Ghosh, Apr 28 2017

Crossrefs

Cf. A007060.

Sequence in context: A296406 A215651 A363180 * A009501 A013027 A012918

Adjacent sequences: A285847 A285848 A285849 * A285851 A285852 A285853

Keyword

nonn,easy

Author

Max Alekseyev, Apr 28 2017

Status

approved