|
%I #29 Aug 05 2020 04:58:35
%S 1,4,96,5760,645120,116121600,30656102400,11158821273600,
%T 5356234211328000,3278015337332736000,2491291656372879360000,
%U 2301953490488540528640000,2541356653499348743618560000
%N a(n) = 2^n*(2*n)!.
%H Harry J. Smith, <a href="/A065140/b065140.txt">Table of n, a(n) for n = 0..100</a>
%F 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)))), n=1, 2,... Integral representation as n-th moment of a positive function on a positive half-axis: a(n)=int(x^n*exp(-sqrt(x/2))/(2*sqrt(2*x)), x=0..infinity), n=0, 1, ....
%F 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 From _Amiram Eldar_, Aug 05 2020: (Start)
%F Sum_{n>=0} 1/a(n) = cosh(sqrt(2)/2).
%F Sum_{n>=0} (-1)^n/a(n) = cos(sqrt(2)/2). (End)
%t Table[2^n (2n)!,{n,0,15}] (* _Harvey P. Dale_, Nov 28 2011 *)
%o (PARI) { for (n=0, 100, write("b065140.txt", n, " ", 2^n*(2*n)!) ) } \\ _Harry J. Smith_, Oct 11 2009
%K nonn
%O 0,2
%A _Karol A. Penson_, Oct 16 2001
|
