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 A010370 a(n) = binomial(2*n, n)^2 / (1-2*n). 7
 1, -4, -12, -80, -700, -7056, -77616, -906048, -11042460, -139053200, -1796567344, -23696871744, -317933029232, -4326899214400, -59605244280000, -829705000377600, -11654762427179100, -165021757273414800, -2353088020380174000, -33764531705178120000 (list; graph; refs; listen; history; text; internal format)
 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: A165261 A027145 A299795 * A197852 A305334 A205337 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

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Last modified January 21 08:09 EST 2020. Contains 331104 sequences. (Running on oeis4.)