A010370 a(n) = binomial(2*n, n)^2 / (1-2*n).

1, -4, -12, -80, -700, -7056, -77616, -906048, -11042460, -139053200, -1796567344, -23696871744, -317933029232, -4326899214400, -59605244280000, -829705000377600, -11654762427179100, -165021757273414800, -2353088020380174000, -33764531705178120000

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

a(n) = A002894(n) - 8 * A000894(n-1). - Michael Somos, Jul 10 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

Cf. A000891, A000894, A002420, A002894

Sequence in context: A027145 A299795 A351736 * A197852 A362384 A305334

Adjacent sequences: A010367 A010368 A010369 * A010371 A010372 A010373

Keyword

sign,easy

Author

Joe Keane (jgk(AT)jgk.org)

Extensions

Additional comments from Michael Somos, Dec 13 2002

Status

approved