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

%I A276178 #21 Dec 18 2018 03:38:58
%S A276178 1,4,12,32,84,240,784,2816,10404,38096,137456,493440,1783376,6532288,
%T A276178 24245568,90814464,341776164,1289126160,4870386736,18439692928,
%U A276178 70004793936,266551445952,1017708956224,3894679004160,14932998810896,57349426579264,220574904103872,849571289810432
%N A276178 G.f.: 1/AGM(1, (1-4*x)^2).
%H A276178 Gheorghe Coserea, <a href="/A276178/b276178.txt">Table of n, a(n) for n = 0..301</a>
%H A276178 Robert S. Maier, <a href="http://arxiv.org/abs/math/0611041">On Rationally Parametrized Modular Equations</a>, arXiv:math/0611041 [math.NT], 2006.
%F A276178 G.f.: 1/agm(1, (1-4*x)^2).
%F A276178 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).
%F A276178 From _Vaclav Kotesovec_, Aug 25 2016: (Start)
%F A276178 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).
%F A276178 a(n) ~ 2^(2*n+2)/(Pi*n).
%F A276178 (End)
%e A276178 A(x) = 1 + 4*x + 12*x^2 + 32*x^3 + ... is the g.f.
%t A276178 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];
%t A276178 Table[a[n], {n, 0, 27}]
%t A276178 (* or: *)
%t A276178 Series[1/FunctionExpand[ArithmeticGeometricMean[1, (1-4x)^2], 1-4x > 0], {x, 0, 28}] // CoefficientList[#, x]& (* _Jean-François Alcover_, Dec 18 2018 *)
%o A276178 (PARI) N=34; x='x + O('x^N); Vec(1/agm(1, (1-4*x)^2))
%Y A276178 Cf. A091401, A276018.
%K A276178 nonn
%O A276178 0,2
%A A276178 _Gheorghe Coserea_, Aug 23 2016