A189849 a(0)=1, a(1)=0, a(n) = 4*n*(n-1)*(a(n-1) + 2*(n-1)*a(n-2)).

1, 0, 16, 384, 23040, 2088960, 278323200, 50969640960, 12290021130240, 3774394191052800, 1438421245702963200, 666120016990568448000, 368420070161105761075200, 239869937154980747988172800, 181598769336835394381021184000, 158184707164826878472739618816000

Offset: 0

Author

Stewart Herring, Apr 29 2011

Keywords

• nonn
• easy

Comments

The number of ways n couples can sit in rows of two seats with no person next to their partner.

a(n)/(2n)! gives the probability of this is and tends to exp(-1/2) as n tends to infinity.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..224

Crossrefs

Cf. A053871, A333706.

Programs

Magma

[(-2)^n*Factorial(n)*(&+[(-1/2)^k*Binomial(n,k)*Factorial(2*k)/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jan 13 2018

Maple

a:= n-> (-2)^n*n!*add((-1/2)^i*binomial(n, i)*(2*i)!/i!, i=0..n): seq(a(n), n=0..20);

Mathematica

Table[(-2)^n*n!*Sum[(-1/2)^i*Binomial[n,i]*(2*i)!/i!,{i,0,n}],{n,1,20}]
RecurrenceTable[{a[0]==1,a[1]==0,a[n]==4n(n-1)(a[n-1]+2(n-1)a[n-2])},a,{n,20}] (* Harvey P. Dale, May 02 2012 *)

Maxima

a[0]:1$ a[1]:0$ a[n]:=4*n*(n-1)*(a[n-1]+2*(n-1)*a[n-2])$ makelist(a[n], n, 0, 13);  /* Bruno Berselli, May 23 2011 */

PARI

for(n=0,30, print1((-2)^n*n!*sum(k=0,n, (-1/2)^k*binomial(n,k)*(2*k)!/k!), ", ")) \\ G. C. Greubel, Jan 13 2018

Formula

a(n) = (-2)^n*n!*hypergeom([ -n, 1/2],[],2).

a(n) = (n!)^2 times the coefficient of x^n in the expansion of exp(-2*x)/sqrt(1-4*x).

a(n) = 2^n*n!*A053871(n).

a(n) = A333706(2n,n). - Alois P. Heinz, Apr 10 2020