A276178 G.f.: 1/AGM(1, (1-4*x)^2).

1, 4, 12, 32, 84, 240, 784, 2816, 10404, 38096, 137456, 493440, 1783376, 6532288, 24245568, 90814464, 341776164, 1289126160, 4870386736, 18439692928, 70004793936, 266551445952, 1017708956224, 3894679004160, 14932998810896, 57349426579264, 220574904103872, 849571289810432

Offset: 0

Author

Gheorghe Coserea, Aug 23 2016

Keywords

• nonn

Examples

			A(x) = 1 + 4*x + 12*x^2 + 32*x^3 + ... is the g.f.
		

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.

Crossrefs

Cf. A091401, A276018.

Programs

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 *)

PARI

N=34; x='x + O('x^N); Vec(1/agm(1, (1-4*x)^2))

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)