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A092266 Expansion of (1+4x)/AGM(1+4x,1-4x) where AGM denotes the arithmetic-geometric mean. 2

%I

%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

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Last modified May 9 23:39 EDT 2021. Contains 343746 sequences. (Running on oeis4.)