A147988 Coefficients of denominator polynomials Q(n,x) associated with reciprocation.

1, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 4, 0, 1, 0, 1, 0, 11, 0, 45, 0, 88, 0, 88, 0, 45, 0, 11, 0, 1, 0, 1, 0, 26, 0, 293, 0, 1896, 0, 7866, 0, 22122, 0, 43488, 0, 60753, 0, 60753, 0, 43488, 0, 22122, 0, 7866, 0, 1896, 0, 293, 0, 26, 0, 1, 0, 1, 0, 57, 0, 1512, 0, 24858, 0, 284578, 0

Offset

1,10

Comments

1. Q(n,1)=A073834(n) for n>=1.

2. For n>=3, Q(n)=Q(n,x)=i*T(n,i*x), where T(n) is the polynomial at A147986.

Thus all the zeros of Q(n,x), for n>=2, are nonreal.

Links

Table of n, a(n) for n=1..73.

Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Formula

The basic idea is to iterate the reciprocation-sum mapping x/y -> x/y+y/x.

Let x be an indeterminate, P(1)=x, Q(1)=1 and for n>1, define P(n)=P(n-1)^2+Q(n-1)^2 and Q(n)=P(n-1)*Q(n-1), so that P(n)/Q(n)=P(n-1)/Q(n-1)-Q(n-1)/P(n-1).

Example

Q(1) = 1
Q(2) = x
Q(3) = x^3+x
Q(4) = x^7+4*x^5+4*x^3+1
so that, as an array, the sequence begins with:
1
1 0
1 0 1 0
1 0 4 0 4 0 1

Crossrefs

Cf. A147985, A147986, A147987, A147989, A147990, A147991, A147992, A147993.

Sequence in context: A170773 A028618 A147986 * A306488 A230031 A019920

Adjacent sequences: A147985 A147986 A147987 * A147989 A147990 A147991

Keyword

nonn,tabf

Author

Clark Kimberling, Nov 24 2008

Status

approved