A109087 Taylor series of a recursively defined function.

1, 1, 2, -1, -2, 1, 5, -2, -16, 16, 32, -62, -3, 11, 57, 806, -2700, -1186, 19710, -28010, -56244, 244815, -169425, -896555, 2589156, -548641, -12351073, 27233521, 13281064, -170962713, 240398832, 486296905, -2098279338, 921544744, 9407106765, -19435065457, -18489392170, 126414945507

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..37.

Formula

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).

Example

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).

Prog

(PARI) N=41; f=x; g=1; for(n=1, N, g/=f; f+=g+O(x^N)); Vec(x*f)

Crossrefs

Sequence in context: A000001 A172133 A146002 * A102048 A318360 A102551

Adjacent sequences: A109084 A109085 A109086 * A109088 A109089 A109090

Keyword

easy,sign

Author

Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Jun 19 2005

Status

approved