OFFSET
0,2
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 0..301
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.
FORMULA
G.f.: 1/agm(1, (1-4*x)^2).
0 = x*(x+2)*(x+4)*(x^2 + 4*x + 8) * y'' + (5*x^4 + 40*x^3 + 120*x^2 + 160*x + 64) * y' + 4*(x+2)^3 * y, where y(x) = A(x/-8).
From Vaclav Kotesovec, Aug 25 2016: (Start)
Recurrence: n^2*a(n) = 2*(5*n^2 - 5*n + 2)*a(n-1) - 8*(5*n^2 - 10*n + 6)*a(n-2) + 16*(5*n^2 - 15*n + 12)*a(n-3) - 64*(n-2)^2*a(n-4).
a(n) ~ 2^(2*n+2)/(Pi*n).
(End)
EXAMPLE
A(x) = 1 + 4*x + 12*x^2 + 32*x^3 + ... is the g.f.
MATHEMATICA
a[n_] = DifferenceRoot[Function[{a, n}, {(-80 n^2 - 400n - 512) a[n+1] + (40n^2 + 240n + 368) a[n+2] + (-10n^2 - 70n - 124) a[n+3] + 64(n+2)^2 a[n] + (n+4)^2 a[n+4] == 0, a[0] == 1, a[1] == 4, a[2] == 12, a[3] == 32}]][n];
Table[a[n], {n, 0, 27}]
(* or: *)
Series[1/FunctionExpand[ArithmeticGeometricMean[1, (1-4x)^2], 1-4x > 0], {x, 0, 28}] // CoefficientList[#, x]& (* Jean-François Alcover, Dec 18 2018 *)
PROG
(PARI) N=34; x='x + O('x^N); Vec(1/agm(1, (1-4*x)^2))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Aug 23 2016
STATUS
approved