A095839 a(n) = ((2*n)!/(n!*2^(n-1)))*integral_{x=1/2..1} (Sqrt(1-x^2)/x)^(2*n) dx.

1, 1, 5, 51, 807, 17445, 479565, 16019955, 630301455, 28552506885, 1463744449125, 83780913568275, 5296205435649975, 366478026602012325, 27552067849812030525, 2236327624673777509875, 194908916445067162713375, 18154937081288124469477125

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..347

Formula

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

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

Maple

A095839 := proc(n)
    local k;
    (4^k-2)/2/(2*k-1) ;
    add(%*(-1)^k*binomial(n, k), k=0..n) ;
    %*(-1)^n*(2*n)!/n!/2^(n-1) ;
end proc: # R. J. Mathar, Feb 13 2014

Mathematica

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 *)
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.

Crossrefs

Sequence in context: A343687 A299435 A377803 * A234290 A107669 A218675

Adjacent sequences: A095836 A095837 A095838 * A095840 A095841 A095842

Keyword

nonn

Author

Al Hakanson (Hawkuu(AT)excite.com), Jun 08 2004

Extensions

a(8)-a(11) from Robert G. Wilson v, Nov 18 2005

Definition corrected by Robert Israel, May 08 2018

Status

approved