%I #5 Aug 28 2018 15:27:05

%S 1,-2,-6,20,-26,324,932,-3864,-12282,-8812,123596,-1011048,1302748,

%T -9066968,-11700216,327436496,500340678,4725531060,3741191612,

%U -11250963784,-147523219212,-1497706973320,-7306482940296,-675852523344,10735087541148

%N Expansion of a parametrization of x^2 - y + y^2 = 0 at x = y = 0.

%F Given g.f. A(x), then B(x) = x * A(x^2 / 4) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v + v^2.

%F Given g.f. A(x), then B(x) = x * A(x^2 / 16) satisfies B(x^2 / 2) = f(B(x)) where f(x) := 1 - sqrt(1 - x^2). Thus, if 0 < x_0 = B(t_0) < 1, x_{n+1} = f(x_n), t_{n+1} = t_n^2 / 2, then x_n = B(t_n). - _Michael Somos_, Aug 28 2018

%e G.f. = 1 - 2*x - 6*x^2 + 20*x^3 - 26*x^4 + 324*x^5 + 932*x^6 - 3864*x^7 + ... - _Michael Somos_, Aug 28 2018

%t a[ n_] := If[ n < 0, 0, 16^n SeriesCoefficient[ Nest[ Sqrt[1 - (1 - (# /. x -> x^2 / 2))^2] &, x, Ceiling[Log[2, n + 1]]], {x, 0, 2 n + 1}]]; (* _Michael Somos_, Aug 28 2018 *)

%o (PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while(m<=n, m*=2; A = subst(A, x, 4*x^2); A = sqrt(A - 4*x*A^2)); polcoeff(A, n))};

%K sign

%O 0,2

%A _Michael Somos_, Mar 24 2005