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A147987
Coefficients of numerator polynomials P(n,x) associated with reciprocation.
5
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
1,8
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.
LINKS
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
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
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 *)
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Nov 24 2008
STATUS
approved