A327834 Expansion of 1 / AGM(1, 1 - 8*x)^2 in powers of x.

1, 8, 56, 384, 2648, 18496, 131008, 940032, 6821848, 49985984, 369258560, 2746629120, 20549693888, 154518118912, 1166873394688, 8844937101312, 67265481552856, 513038965707968, 3923108472072512, 30068733313938432, 230943237733355840, 1777114026405752320

Offset

0,2

Comments

AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Links

Table of n, a(n) for n=0..21.

Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean

Formula

Recurrence: n^3*a(n) = 4*(2*n - 1)*(3*n^2 - 3*n + 2)*a(n-1) - 16*(n-1)*(13*n^2 - 26*n + 20)*a(n-2) + 128*(2*n - 3)*(3*n^2 - 9*n + 8)*a(n-3) - 1024*(n-2)^3*a(n-4).

a(n) ~ 2^(3*n + 3) * (log(4*n) + gamma) / (Pi^2 * n), where gamma is the Euler-Mascheroni constant A001620.

Mathematica

CoefficientList[Series[(2*EllipticK[1/(1 - 1/(4*x))^2]/(Pi*(1 - 4*x)))^2, {x, 0, 25}], x]
CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16*x*(1 - 4*x)]^2, {x, 0, 25}], x]

Crossrefs

Cf. A081085, A089603.

Sequence in context: A003494 A057084 A101596 * A092521 A156088 A002914

Adjacent sequences: A327831 A327832 A327833 * A327835 A327836 A327837

Keyword

nonn

Author

Vaclav Kotesovec, Sep 27 2019

Status

approved