OFFSET
0,2
COMMENTS
Expansion of hypergeometric function F(-1/2, 1/2; 1; 16*x).
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
REFERENCES
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.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
LINKS
Robert Israel, Table of n, a(n) for n = 0..835
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
a(n) ~ -1/2*Pi^-1*n^-2*2^(4*n). [corrected by Vaclav Kotesovec, Oct 04 2019]
a(n) = -4 * A000891(n-1), n>0. - Michael Somos, Dec 13 2002
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
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
n^2*a(n) -4*(2*n-1)*(2*n-3)*a(n-1)=0. - R. J. Mathar, Feb 15 2013
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
EXAMPLE
G.f. = 1 - 4*x - 12*x^2 - 80*x^3 - 700*x^4 - 7056*x^5 - 77616*x^6 - ...
MAPLE
seq(binomial(2*n, n)^2/(1-2*n), n=0..30); # Robert Israel, Jul 10 2017
MATHEMATICA
CoefficientList[Series[EllipticE[16x]2/Pi, {x, 0, 20}], x]
Table[Binomial[2n, n]^2/(1-2n), {n, 0, 30}] (* Harvey P. Dale, Mar 07 2013 *)
PROG
(PARI) {a(n) = binomial(2*n, n)^2 / (1 - 2*n)}; /* Michael Somos, Dec 13 2002 */
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Joe Keane (jgk(AT)jgk.org)
EXTENSIONS
Additional comments from Michael Somos, Dec 13 2002
STATUS
approved