A065140 a(n) = 2^n*(2*n)!.

1, 4, 96, 5760, 645120, 116121600, 30656102400, 11158821273600, 5356234211328000, 3278015337332736000, 2491291656372879360000, 2301953490488540528640000, 2541356653499348743618560000, 3303763649549153366704128000000, 4995290638118319890456641536000000

Offset: 0

Author

Karol A. Penson, Oct 16 2001

Keywords

• nonn
• easy
• changed

Comments

From Enrique Navarrete, Aug 29 2025: (Start)

For n > 0, 1/2*a(n) is the number of ways to seat 2*n people on linearly ordered benches placing an even number of people (>=2) on each bench.

For example, 1/2*a(4)=322560 since the number of ways are (number of people in parentheses):

1 bench (8): 40320 ways;

2 benches (6,2): 80640 ways;

2 benches (4,4): 40320 ways;

3 benches (4,2,2): 120960 ways;

4 benches (2,2,2,2): 40320 ways.

If the benches are not linearly ordered the number of ways is A088026.

If we seat an odd number of people on linearly ordered benches the number of ways is A005443. (End)

Links

• Harry J. Smith, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000680, A088026.

Programs

Mathematica

Table[2^n (2n)!,{n,0,15}] (* Harvey P. Dale, Nov 28 2011 *)

PARI

{ for (n=0, 100, write("b065140.txt", n, " ", 2^n*(2*n)!) ) } \\ Harry J. Smith, Oct 11 2009

Formula

Hypergeometric generating function, in Maple notation: 1/sqrt(1-8*x), i.e., a(0)=1, a(n)=round(evalf(subs(x=0, n!*diff(1/(sqrt(1-8*x)), x$n)))), for n>=1.

Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x>=0} x^n*exp(-sqrt(x/2))/(2*sqrt(2*x)) dx, for n>=0.

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 4*x*(k+1)*(2*k+1)/(4*x*(k+1)*(2*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013

From Amiram Eldar, Aug 05 2020: (Start)

Sum_{n>=0} 1/a(n) = cosh(sqrt(2)/2).

Sum_{n>=0} (-1)^n/a(n) = cos(sqrt(2)/2). (End)

From Alexandre Herrera, Apr 18 2025: (Start)

Sum_{n>=0} x^(4*n)*(-1)^(n)/a(2n) = cos(x/2)*cosh(x/2).

Sum_{n>=0} x^(4*n+2)*(-1)^(n)/a(2n+1) = sin(x/2)*sinh(x/2).

Sum_{n>=0} x^(2*n)*(-1)^(n)/a(n) = cos(x*sqrt(2)/2).

Sum_{n>=0} x^(2*n)/a(n) = cosh(x*sqrt(2)/2). (End)