This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A065140 #55 Sep 03 2025 19:49:01
%S A065140 1,4,96,5760,645120,116121600,30656102400,11158821273600,
%T A065140 5356234211328000,3278015337332736000,2491291656372879360000,
%U A065140 2301953490488540528640000,2541356653499348743618560000,3303763649549153366704128000000,4995290638118319890456641536000000
%N A065140 a(n) = 2^n*(2*n)!.
%C A065140 From _Enrique Navarrete_, Aug 29 2025: (Start)
%C A065140 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.
%C A065140 For example, 1/2*a(4)=322560 since the number of ways are (number of people in parentheses):
%C A065140 1 bench (8): 40320 ways;
%C A065140 2 benches (6,2): 80640 ways;
%C A065140 2 benches (4,4): 40320 ways;
%C A065140 3 benches (4,2,2): 120960 ways;
%C A065140 4 benches (2,2,2,2): 40320 ways.
%C A065140 If the benches are not linearly ordered the number of ways is A088026.
%C A065140 If we seat an odd number of people on linearly ordered benches the number of ways is A005443. (End)
%H A065140 Harry J. Smith, <a href="/A065140/b065140.txt">Table of n, a(n) for n = 0..100</a>
%F A065140 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.
%F A065140 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.
%F A065140 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
%F A065140 From _Amiram Eldar_, Aug 05 2020: (Start)
%F A065140 Sum_{n>=0} 1/a(n) = cosh(sqrt(2)/2).
%F A065140 Sum_{n>=0} (-1)^n/a(n) = cos(sqrt(2)/2). (End)
%F A065140 From _Alexandre Herrera_, Apr 18 2025: (Start)
%F A065140 Sum_{n>=0} x^(4*n)*(-1)^(n)/a(2n) = cos(x/2)*cosh(x/2).
%F A065140 Sum_{n>=0} x^(4*n+2)*(-1)^(n)/a(2n+1) = sin(x/2)*sinh(x/2).
%F A065140 Sum_{n>=0} x^(2*n)*(-1)^(n)/a(n) = cos(x*sqrt(2)/2).
%F A065140 Sum_{n>=0} x^(2*n)/a(n) = cosh(x*sqrt(2)/2). (End)
%t A065140 Table[2^n (2n)!,{n,0,15}] (* _Harvey P. Dale_, Nov 28 2011 *)
%o A065140 (PARI) { for (n=0, 100, write("b065140.txt", n, " ", 2^n*(2*n)!) ) } \\ _Harry J. Smith_, Oct 11 2009
%Y A065140 Cf. A000680, A088026.
%K A065140 nonn,easy,changed
%O A065140 0,2
%A A065140 _Karol A. Penson_, Oct 16 2001