A104749 Expansion of a parametrization of x^2 - y + y^2 = 0 at x = y = 0.

1, -2, -6, 20, -26, 324, 932, -3864, -12282, -8812, 123596, -1011048, 1302748, -9066968, -11700216, 327436496, 500340678, 4725531060, 3741191612, -11250963784, -147523219212, -1497706973320, -7306482940296, -675852523344, 10735087541148

Offset

0,2

Links

Table of n, a(n) for n=0..24.

Formula

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.

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

Example

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

Mathematica

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 *)

Prog

(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))};

Crossrefs

Sequence in context: A343506 A128447 A032622 * A062281 A267124 A322371

Adjacent sequences: A104746 A104747 A104748 * A104750 A104751 A104752

Keyword

sign

Author

Michael Somos, Mar 24 2005

Status

approved