A089624 - OEIS

%I #21 Sep 29 2019 08:59:55

%S 1,2,16,168,1986,25092,330816,4492560,62352720,879956000,12583279360,

%T 181872982400,2652039363240,38959845007440,575974743052800,

%U 8561706637619520,127874111328349890,1917875205285147780

%N Expansion of sqrt(2/Pi*EllipticK(4*sqrt(x))).

%C When convolved with itself gives A002894.

%H G. C. Greubel, <a href="/A089624/b089624.txt">Table of n, a(n) for n = 0..450</a>

%F 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

%F 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

%t 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 *)

%t nmax = 20; CoefficientList[Series[Sqrt[2*EllipticK[16*x]/Pi], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 10 2018 *)

%o (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 */

%o (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 */

%Y Cf. A036917.

%K nonn

%O 0,2

%A D. G. Rogers and _Vladeta Jovovic_, Dec 31 2003