%I A010370
%S A010370 1,4,12,80,700,7056,77616,906048,11042460,139053200,1796567344,
%T A010370 23696871744,317933029232,4326899214400,59605244280000,829705000377600,
%U A010370 11654762427179100,165021757273414800,2353088020380174000
%V A010370 1,-4,-12,-80,-700,-7056,-77616,-906048,-11042460,-139053200,-1796567344,
%W A010370 -23696871744,-317933029232,-4326899214400,-59605244280000,-829705000377600,
%X A010370 -11654762427179100,-165021757273414800,-2353088020380174000
%N A010370 C(2*n,n)^2 / (1-2*n).
%C A010370 Expansion of hypergeometric function F(-1/2,1/2;1;16x).
%C A010370 Expansion of E(m)/(pi/2) in powers of m/16=(k/4)^2, where E(m) is complete 
               elliptic integral of second kind evaluated at m. - Michael Somos, 
               Mar 04 2003
%D A010370 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 A010370 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
%H A010370 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%F A010370 a(n) ~ 1/2*pi^-1*n^-2*2^(4*n)
%F A010370 G.f.: F(-1/2, 1/2;1;16x) = E(16x)/(pi/2). a(n)=C(2*n, n)^2/(1-2*n). - 
               Michael Somos, Mar 04 2003
%F A010370 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
%t A010370 CoefficientList[Series[EllipticE[16x]2/Pi, {x, 0, 20}], x]
%o A010370 (PARI) a(n)=if(n<0,0,binomial(2*n,n)^2/(1-2*n))
%Y A010370 Cf. A002894, A002420. a(n)=-4*A000891(n-1), n>0.
%Y A010370 Sequence in context: A078628 A165261 A027145 this_sequence A081214 A064280 
               A096424
%Y A010370 Adjacent sequences: A010367 A010368 A010369 this_sequence A010371 A010372 
               A010373
%K A010370 sign,easy
%O A010370 0,2
%A A010370 Joe Keane (jgk(AT)jgk.org)
%E A010370 Additional comments from Michael Somos, Dec 13 2002