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%I #2 Feb 27 2009 03:00:00
%S 1,1,2,-1,-2,1,5,-2,-16,16,32,-62,-3,11,57,806,-2700,-1186,19710,
%T -28010,-56244,244815,-169425,-896555,2589156,-548641,-12351073,
%U 27233521,13281064,-170962713,240398832,486296905,-2098279338,921544744,9407106765,-19435065457,-18489392170,126414945507
%N Taylor series of a recursively defined function.
%C With polynomials p_n(x) from A109086, g_n(x) = prod(j = 0, n-3)p_j(x)^(n-j-2)/p_(n-1)(x), f_n(x) = p_n(x)/prod(j = 0, n-1)p_j(x), n >= 0. Taylor series of 1/f(x) is A109088. f(1) = f(2) = A109089(decimal expansion) = A109090(continued fraction). The function f(x) has poles at 0, i and -i, a real minimum at about 1.448 and a real maximum at about -0.904.
%F Define two sequences of rational functions g_0(x) = 1, f_0(x) = x, g_(n+1)(x) = g_n(x)/f_n(x), f_(n+1)(x) = f_n(x)+g_n(x), n >= 0. Then define the function f(x) = lim(n->infinity)f_n(x), sum(n = 0, infinity)a(n)x^n = x*f(x).
%e x*f(x) = 1 + x + 2*x^2 - x^3 - 2*x^4 + x^5 + 5*x^6 - 2*x^7 - 16*x^8 + 16*x^9 + 32*x ^10 - 62*x^11 - 3*x^12 + 11*x^13 + 57*x^14 + 806*x^15 + O(x^16).
%o (PARI) N=41;f=x;g=1;for(n=1,N,g/=f;f+=g+O(x^N));Vec(x*f)
%K easy,sign
%O 0,3
%A Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Jun 19 2005
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