%I A109087
%S A109087 1,1,2,1,2,1,5,2,16,16,32,62,3,11,57,806,2700,1186,19710,28010,56244,244815,
%T A109087 169425,896555,2589156,548641,12351073,27233521,13281064,170962713,240398832,
%U A109087 486296905,2098279338,921544744,9407106765,19435065457,18489392170,126414945507
%V A109087 1,1,2,-1,-2,1,5,-2,-16,16,32,-62,-3,11,57,806,-2700,-1186,19710,-28010,
               -56244,244815,
%W A109087 -169425,-896555,2589156,-548641,-12351073,27233521,13281064,-170962713,
               240398832,
%X A109087 486296905,-2098279338,921544744,9407106765,-19435065457,-18489392170,
               126414945507
%N A109087 Taylor series of a recursively defined function.
%C A109087 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 A109087 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 A109087 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 A109087 (PARI) N=41;f=x;g=1;for(n=1,N,g/=f;f+=g+O(x^N));Vec(x*f)
%Y A109087 Sequence in context: A128644 A000001 A146002 this_sequence A102048 A102551 
               A152823
%Y A109087 Adjacent sequences: A109084 A109085 A109086 this_sequence A109088 A109089 
               A109090
%K A109087 easy,sign
%O A109087 0,3
%A A109087 Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Jun 19 2005