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%I #36 Oct 04 2019 13:15:43
%S 1,-4,-12,-80,-700,-7056,-77616,-906048,-11042460,-139053200,
%T -1796567344,-23696871744,-317933029232,-4326899214400,
%U -59605244280000,-829705000377600,-11654762427179100,-165021757273414800,-2353088020380174000,-33764531705178120000
%N a(n) = binomial(2*n, n)^2 / (1-2*n).
%C Expansion of hypergeometric function F(-1/2, 1/2; 1; 16*x).
%C Expansion of E(m)/(Pi/2) in powers of m/16 = (k/4)^2, where E(m) is the complete elliptic integral of the second kind evaluated at m. - _Michael Somos_, Mar 04 2003
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 591.
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
%H Robert Israel, <a href="/A010370/b010370.txt">Table of n, a(n) for n = 0..835</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F a(n) ~ -1/2*Pi^-1*n^-2*2^(4*n). [corrected by _Vaclav Kotesovec_, Oct 04 2019]
%F a(n) = -4 * A000891(n-1), n>0. - _Michael Somos_, Dec 13 2002
%F G.f.: F(-1/2, 1/2; 1; 16x) = E(16*x) / (Pi/2). a(n) = binomial(2*n, n)^2 / (1 - 2*n). - _Michael Somos_, Mar 04 2003
%F E.g.f.: Sum_{n>=0} a(n) * (x/2)^(2n)/(2n)! = I0^2*(1-2*x^2) +2*x*I0*I1 +2*x^2*I1^2 where I0=BesselI(0, x), I1=BesselI(1, x). - _Michael Somos_, Jun 22 2005
%F n^2*a(n) -4*(2*n-1)*(2*n-3)*a(n-1)=0. - _R. J. Mathar_, Feb 15 2013
%F 0 = a(n)*(+1048576*a(n+2) + 2695168*a(n+3) - 989568*a(n+4) + 65340*a(n+5)) + a(n+1)*(-8192*a(n+2) - 99840*a(n+3) + 52652*a(n+4) - 4236*a(n+5)) + a(n+2)*(-128*a(n+2) + 280*a(n+3) - 484*a(n+4) + 57*a(n+5)) for all n in Z. - _Michael Somos_, Jan 21 2017
%F a(n) = A002894(n) - 8 * A000894(n-1). - _Michael Somos_, Jul 10 2017
%e G.f. = 1 - 4*x - 12*x^2 - 80*x^3 - 700*x^4 - 7056*x^5 - 77616*x^6 - ...
%p seq(binomial(2*n,n)^2/(1-2*n), n=0..30); # _Robert Israel_, Jul 10 2017
%t CoefficientList[Series[EllipticE[16x]2/Pi, {x, 0, 20}], x]
%t Table[Binomial[2n,n]^2/(1-2n),{n,0,30}] (* _Harvey P. Dale_, Mar 07 2013 *)
%o (PARI) {a(n) = binomial(2*n, n)^2 / (1 - 2*n)}; /* _Michael Somos_, Dec 13 2002 */
%Y Cf. A000891, A000894, A002420, A002894
%K sign,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
%E Additional comments from _Michael Somos_, Dec 13 2002
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