|
%I #16 May 08 2018 21:01:17
%S 1,1,5,51,807,17445,479565,16019955,630301455,28552506885,
%T 1463744449125,83780913568275,5296205435649975,366478026602012325,
%U 27552067849812030525,2236327624673777509875,194908916445067162713375,18154937081288124469477125
%N a(n) = ((2*n)!/(n!*2^(n-1)))*integral_{x=1/2..1} (Sqrt(1-x^2)/x)^(2*n) dx.
%H Robert Israel, <a href="/A095839/b095839.txt">Table of n, a(n) for n = 0..347</a>
%F Conjecture: a(n) +(-4*n+9)*a(n-1) -6*(n-1)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Feb 13 2014
%F E.g.f.: (2-sqrt(1-6*x))/(1+2*x). Conjecture follows from the d.e. (12*x^2+4*x-1)*y''+(30*x-1)*y'+6*y=0 satisfied by this. - _Robert Israel_, May 08 2018
%p A095839 := proc(n)
%p local k;
%p (4^k-2)/2/(2*k-1) ;
%p add(%*(-1)^k*binomial(n,k),k=0..n) ;
%p %*(-1)^n*(2*n)!/n!/2^(n-1) ;
%p end proc: # _R. J. Mathar_, Feb 13 2014
%t f[n_] := Numerator[ Integrate[(Sqrt[1 - x^2]/x)^(2n), {x, 1/2, 1}]*(2n)!/(n!2^(n + 1)!)]; Table[ f[n], {n, 0, 11}] (* _Robert G. Wilson v_ *)
%t Numerator[2^(-2 - Gamma[2 + n])*3^(1 + n)*(2*n)!* Hypergeometric2F1Regularized[1, 1/2 + n, 2 + n, -3]] _Eric W. Weisstein_, Nov 19 2005.
%K nonn
%O 0,3
%A Al Hakanson (Hawkuu(AT)excite.com), Jun 08 2004
%E a(8)-a(11) from _Robert G. Wilson v_, Nov 18 2005
%E Definition corrected by _Robert Israel_, May 08 2018
| 