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A323385 Expansion of AGM(1,1-16*x) (where AGM denotes the arithmetic-geometric mean).

%I A323385 #21 Feb 16 2025 08:33:57
%S A323385 1,-8,-16,-128,-1344,-15872,-199680,-2613248,-35148800,-482500608,
%T A323385 -6730072064,-95094702080,-1358152794112,-19573573681152,
%U A323385 -284284750397440,-4156674357067776,-61133523873169408,-903754859816157184,-13421680957337894912,-200140704802846801920
%N A323385 Expansion of AGM(1,1-16*x) (where AGM denotes the arithmetic-geometric mean).
%H A323385 Robert Israel, <a href="/A323385/b323385.txt">Table of n, a(n) for n = 0..833</a>
%H A323385 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Arithmetic-GeometricMean.html">Arithmetic-geometric mean</a>
%H A323385 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean">Arithmetic-geometric mean</a>
%F A323385 a(n) = 2^n * A060691(n).
%F A323385 a(n) ~ -Pi * 2^(4*n-1) / (n * log(n)^2) * (1 - (2*gamma + 4*log(2))/log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 29 2019
%p A323385 R:= Pi*(1-8*x)/(2*EllipticK(-8*x/(1-8*x))):
%p A323385 S:= series(R,x,31):
%p A323385 seq(coeff(S,x,j),j=0..30); # _Robert Israel_, Jan 13 2019
%t A323385 CoefficientList[Series[Pi*(1 - 16*x) / (2*EllipticK[1 - 1/(1 - 16*x)^2]), {x, 0, 25}], x] (* _Vaclav Kotesovec_, Sep 28 2019 *)
%o A323385 (PARI) N=66; x='x+O('x^N); Vec(agm(1, 1-16*x))
%Y A323385 Convolution inverse of A053175.
%Y A323385 Cf. A060691, A089602, A090004.
%K A323385 sign
%O A323385 0,2
%A A323385 _Seiichi Manyama_, Jan 13 2019