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Search: id:A010370
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| 1, -4, -12, -80, -700, -7056, -77616, -906048, -11042460, -139053200, -1796567344, -23696871744, -317933029232, -4326899214400, -59605244280000, -829705000377600, -11654762427179100, -165021757273414800, -2353088020380174000
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Expansion of hypergeometric function F(-1/2,1/2;1;16x).
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
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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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a(n) ~ 1/2*pi^-1*n^-2*2^(4*n)
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
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
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MATHEMATICA
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CoefficientList[Series[EllipticE[16x]2/Pi, {x, 0, 20}], x]
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PROGRAM
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(PARI) a(n)=if(n<0, 0, binomial(2*n, n)^2/(1-2*n))
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CROSSREFS
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Cf. A002894, A002420. a(n)=-4*A000891(n-1), n>0.
Sequence in context: A078628 A165261 A027145 this_sequence A081214 A064280 A096424
Adjacent sequences: A010367 A010368 A010369 this_sequence A010371 A010372 A010373
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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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Additional comments from Michael Somos, Dec 13 2002
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