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
Examples
G.f. = 1 - 4*x - 12*x^2 - 80*x^3 - 700*x^4 - 7056*x^5 - 77616*x^6 - ...
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].
Programs
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Maple
seq(binomial(2*n,n)^2/(1-2*n), n=0..30); # Robert Israel, Jul 10 2017
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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 *) -
PARI
{a(n) = binomial(2*n, n)^2 / (1 - 2*n)}; /* Michael Somos, Dec 13 2002 */
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
Extensions
Additional comments from Michael Somos, Dec 13 2002
Comments