A089624 Expansion of sqrt(2/Pi*EllipticK(4*sqrt(x))).
1, 2, 16, 168, 1986, 25092, 330816, 4492560, 62352720, 879956000, 12583279360, 181872982400, 2652039363240, 38959845007440, 575974743052800, 8561706637619520, 127874111328349890, 1917875205285147780
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..450
Crossrefs
Cf. A036917.
Programs
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Mathematica
nmax = 20; CoefficientList[Series[Sqrt[Sum[Binomial[2*k, k]^2*x^k, {k, 0, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *) nmax = 20; CoefficientList[Series[Sqrt[2*EllipticK[16*x]/Pi], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *) -
PARI
{a(n) = if(n<0, 0, polcoeff( sqrt( sum(k=0, n, binomial(2*k, k)^2 * x^k, x*O(x^n)) ), n))} /* Michael Somos, Aug 17 2007 */ -
PARI
{a(n) = local(A); if(n<0, 0, A = x*O(x^n); polcoeff( subst( sum(k = 1, sqrtint(n), 2*x^k^2, 1+A), x, serreverse(x * (eta(x+A) * eta(x^4+A)^2 / eta(x^2+A)^3)^8 )), n))} /* Michael Somos, Aug 17 2007 */
Formula
Expansion of theta_3(q) in powers of (m/16) where q = exp(-Pi K'/K) and m = k^2 is the elliptic modulus. - Michael Somos, Aug 17 2007
a(n) ~ 2^(4*n-1) / (n*sqrt(Pi*log(n))) * (1 - (gamma/2 + 2*log(2)) / log(n) + (3*gamma^2/8 + 3*log(2)*gamma + 6*log(2)^2 - Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019
Comments