A147987 Coefficients of numerator polynomials P(n,x) associated with reciprocation.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 7, 0, 13, 0, 7, 0, 1, 1, 0, 15, 0, 83, 0, 220, 0, 303, 0, 220, 0, 83, 0, 15, 0, 1, 1, 0, 31, 0, 413, 0, 3141, 0, 15261, 0, 50187, 0, 115410, 0, 189036, 0, 222621, 0, 189036, 0, 115410, 0, 50187, 0, 15261, 0, 3141, 0, 413, 0, 31, 0, 1, 1, 0, 63

Offset: 1

Author

Clark Kimberling, Nov 24 2008

Keywords

• nonn
• tabf

Comments

1. P(n,1)=A073833(n) for n>=1; P(n,2)=A073833(n+1) for n>=0.

2. P(n)=P(n-1)^2+P(n-1)*P(n-2)^2-P(n-2)^4 for n>=3.

3. For n>=3, P(n)=P(n,x)=S(n,i*x), where S(n) is the polynomial at A147985.

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

Examples

			P(1) = x
P(2) = x^2+1
P(3) = x^4+3*x^2+1
P(4) = x^8+7*x^6+13*x^4+7x^2+1
so that, as an array, the sequence begins with:
1 0
1 0 1
1 0 3 0 1
1 0 7 0 13 0 7 0 1
		

Links

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

Crossrefs

Cf. A147985, A147986, A147988, A147989, A147990, A147991, A147992, A147993.

Programs

Mathematica

p[1] = x; q[1] = 1; p[n_] := p[n] = p[n-1]^2 + q[n-1]^2; q[n_] := q[n] = p[n-1]*q[n-1]; row[n_] := CoefficientList[p[n], x] // Reverse; Table[row[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Apr 22 2013 *)

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).