|
%I #34 Feb 06 2023 07:25:22
%S 1,0,16,384,23040,2088960,278323200,50969640960,12290021130240,
%T 3774394191052800,1438421245702963200,666120016990568448000,
%U 368420070161105761075200,239869937154980747988172800,181598769336835394381021184000,158184707164826878472739618816000
%N a(0)=1, a(1)=0, a(n) = 4*n*(n-1)*(a(n-1) + 2*(n-1)*a(n-2)).
%C The number of ways n couples can sit in rows of two seats with no person next to their partner.
%C a(n)/(2n)! gives the probability of this is and tends to exp(-1/2) as n tends to infinity.
%H G. C. Greubel, <a href="/A189849/b189849.txt">Table of n, a(n) for n = 0..224</a>
%F a(n) = (-2)^n*n!*hypergeom([ -n, 1/2],[],2).
%F a(n) = (n!)^2 times the coefficient of x^n in the expansion of exp(-2*x)/sqrt(1-4*x).
%F a(n) = 2^n*n!*A053871(n).
%F a(n) = A333706(2n,n). - _Alois P. Heinz_, Apr 10 2020
%p a:= n-> (-2)^n*n!*add((-1/2)^i*binomial(n, i)*(2*i)!/i!, i=0..n): seq(a(n), n=0..20);
%t Table[(-2)^n*n!*Sum[(-1/2)^i*Binomial[n,i]*(2*i)!/i!,{i,0,n}],{n,1,20}]
%t 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 *)
%o (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 */
%o (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
%o (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
%Y Cf. A053871, A333706.
%K nonn,easy
%O 0,3
%A _Stewart Herring_, Apr 29 2011
|
