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

1, -8, -16, -128, -1344, -15872, -199680, -2613248, -35148800, -482500608, -6730072064, -95094702080, -1358152794112, -19573573681152, -284284750397440, -4156674357067776, -61133523873169408, -903754859816157184, -13421680957337894912, -200140704802846801920

Offset: 0

Author

Seiichi Manyama, Jan 13 2019

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..833
• Eric Weisstein's World of Mathematics, Arithmetic-geometric mean
• Wikipedia, Arithmetic-geometric mean

Crossrefs

Convolution inverse of A053175.

Cf. A060691, A089602, A090004.

Programs

Maple

R:= Pi*(1-8*x)/(2*EllipticK(-8*x/(1-8*x))):
S:= series(R,x,31):
seq(coeff(S,x,j),j=0..30); # Robert Israel, Jan 13 2019

Mathematica

CoefficientList[Series[Pi*(1 - 16*x) / (2*EllipticK[1 - 1/(1 - 16*x)^2]), {x, 0, 25}], x] (* Vaclav Kotesovec, Sep 28 2019 *)

PARI

N=66; x='x+O('x^N); Vec(agm(1, 1-16*x))

Formula

a(n) = 2^n * A060691(n).

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