Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #12 Sep 27 2019 12:13:32
%S 1,4,4,16,36,144,400,1600,4900,19600,63504,254016,853776,3415104,
%T 11778624,47114496,165636900,662547600,2363904400,9455617600,
%U 34134779536,136539118144,497634306624,1990537226496,7312459672336
%N Expansion of (1+4x)/AGM(1+4x,1-4x) where AGM denotes the arithmetic-geometric mean.
%F G.f.: (1+4x)/AGM(1+4x, 1-4x) where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre.
%F a(n) = A063886(n)^2.
%F a(2n) = A002894(n); a(2n+1) = 4*a(2n).
%F a(n) ~ 2^(2*n + 1) / (Pi*n). - _Vaclav Kotesovec_, Sep 27 2019
%t CoefficientList[Series[2*(1 + 4*x)*EllipticK[1 - (1 + 4*x)^2/(1 - 4*x)^2] / (Pi*(1 - 4*x)), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Sep 27 2019 *)
%o (PARI) a(n)=((n==0)+2*binomial(n-1,(n-1)\2))^2;
%o (PARI) Vec( 1/agm(1,(1-4*x)/(1+4*x)+O(x^66)) ) \\ _Joerg Arndt_, Aug 14 2013
%K nonn
%O 0,2
%A _Michael Somos_, Feb 16 2004