This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369539 #11 Jan 29 2024 20:44:51
%S A369539 1,8,160,4192,125184,4039264,137183488,4831873408,174884458496,
%T A369539 6464875435872,243049515606272,9264347436276608,357204831146577920,
%U A369539 13906950967902306944,545951685104975276032,21587442538147647608320,858975581766808512823296,34369283236381014527279456
%N A369539 Expansion of g.f. A(x) satisfying A(x) = 1 + 8*x * AGM(A(x)^2, A(x)^3).
%C A369539 Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
%H A369539 Paul D. Hanna, <a href="/A369539/b369539.txt">Table of n, a(n) for n = 0..400</a>
%F A369539 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A369539 (1) A(x) = 1 + 8*x * AGM(A(x)^2, A(x)^3).
%F A369539 (2) A(x) = 1 + 8*x * AGM(A(x)^(5/2), (A(x)^2 + A(x)^3)/2).
%F A369539 (3) A(x) = 1 + 8 * Series_Reversion( x / AGM((1 + 8*x)^2, (1 + 8*x)^3) ).
%F A369539 (4) A( x / AGM((1 + 8*x)^2, (1 + 8*x)^3) ) = 1 + 8*x.
%F A369539 a(n) ~ c * d^n / n^(3/2), where d = 43.7139872016060880921082193574226064477439580563964019841877818207326... and c = 0.32250297108028000960144303111184352981179935271075437927423118550208... - _Vaclav Kotesovec_, Jan 29 2024
%F A369539 A(1/d) = 1.6405711647668295617017794194853407... where d is given above. - _Paul D. Hanna_, Jan 29 2024
%e A369539 G.f.: A(x) = 1 + 8*x + 160*x^2 + 4192*x^3 + 125184*x^4 + 4039264*x^5 + 137183488*x^6 + 4831873408*x^7 + 174884458496*x^8 + 6464875435872*x^9 + 243049515606272*x^10 + ...
%e A369539 RELATED SERIES.
%e A369539 x/AGM((1 + 8*x)^2, (1 + 8*x)^3) = x - 20*x^2 + 276*x^3 - 3248*x^4 + 34980*x^5 - 356112*x^6 + 3487568*x^7 - 33204160*x^8 + 309415716*x^9 - 2835178320*x^10 + ...
%e A369539 where A( x/AGM((1 + 8*x)^2, (1 + 8*x)^3) ) = 1 + 8*x.
%t A369539 (* Calculation of constants {d,c}: *) {1/r, s*(s - 1)*Sqrt[(1 + s)/(2*Pi*(4 + s - 7*s^2 + 4*s^3))]} /. FindRoot[{1 + 4*Pi*r*s^3/EllipticK[1 - 1/s^2] == s, 4*Pi*r*s*(2 + s - 2*s^2) + (-1 + s)*EllipticE[1 - 1/s^2] == 0}, {r, 1/50}, {s, 3/2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 29 2024 *)
%o A369539 (PARI) /* From definition: A(x) = 1 + 8*x*AGM(A(x)^2, A(x)^3) */
%o A369539 {a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A^2, A^3)); polcoeff(A, n)}
%o A369539 for(n=0, 30, print1(a(n), ", "))
%o A369539 (PARI) /* From formula: A(x) = 1 + 8*x*AGM(A(x)^(5/2), (A(x)^2 + A(x)^3)/2) */
%o A369539 {a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A^(5/2), (A^2 + A^3)/2)); polcoeff(A, n)}
%o A369539 for(n=0, 30, print1(a(n), ", "))
%o A369539 (PARI) /* From A(x) = 1 + 8*Series_Reversion(x/AGM((1+8*x)^2, (1+8*x)^3)) */
%o A369539 {a(n) = my(A=1); A = 1 + 8*serreverse(x/agm((1+8*x)^2, (1+8*x)^3 +x*O(x^n))); polcoeff(A, n)}
%o A369539 for(n=0, 30, print1(a(n), ", "))
%Y A369539 Cf. A171454, A272823, A369536, A369537, A369538.
%K A369539 nonn
%O A369539 0,2
%A A369539 _Paul D. Hanna_, Jan 28 2024