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A010370
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a(n) = binomial(2*n, n)^2 / (1-2*n).
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7
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1, -4, -12, -80, -700, -7056, -77616, -906048, -11042460, -139053200, -1796567344, -23696871744, -317933029232, -4326899214400, -59605244280000, -829705000377600, -11654762427179100, -165021757273414800, -2353088020380174000, -33764531705178120000
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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].
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FORMULA
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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
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EXAMPLE
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G.f. = 1 - 4*x - 12*x^2 - 80*x^3 - 700*x^4 - 7056*x^5 - 77616*x^6 - ...
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MAPLE
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seq(binomial(2*n, n)^2/(1-2*n), n=0..30); # Robert Israel, Jul 10 2017
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n) = binomial(2*n, n)^2 / (1 - 2*n)}; /* Michael Somos, Dec 13 2002 */
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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Joe Keane (jgk(AT)jgk.org)
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EXTENSIONS
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STATUS
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approved
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