A147986 -id:A147986 - cp's OEIS frontend

A147985 Coefficients of numerator polynomials S(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

Offset: 1

Clark Kimberling, Nov 24 2008

S(n)=U(n-1)V(n-1) where U(n-1)=S(n-1)+S(1)*S(2)*...*S(n-2) and V(n-1)=S(n-1)-S(1)*S(2)*...*S(n-2), for n>=2. If U(n) and V(n) are written as polynomials U(n,x) and V(n,x), then V(n,x)=U(n,-x). See A147989 for coefficients of U(n).
S(n)=S(n-1)^2+S(n-1)*S(n-2)^2-S(n-2)^4 for n>2. (The Gorskov-Wirsting polynomials also have this recurrence; see H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, 84, AMS, pp. 183-190.)
For n>0, the 2^(n-1) zeros of S(n) are real. If r is a zero of S(n), then -r and 1/r are zeros of S(n).
If r is a zero of S(n), then the numbers z satisfying r=z-1/z and r=z+1/z are zeros of S(n+1).
If n>2, then S(n,1)=1 and S(n,2)=A127814(n).
S(n,2^(1/2))=-1 for n>2 and S(n,2^(-1/2))=-2^(1-n) for n>1.

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

Peter J. C. Moses, Rows n = 1..13 of irregular triangle, flattened
Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Cf. A147986, A147987, A147988, A147989, A147990, A147991, A147992, A147993.

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

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

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

A147989 Coefficients of factor polynomials U(n,x) associated with reciprocation.

Original entry on oeis.org

1, 1, -1, 1, 1, -3, -1, 1, 1, 1, -7, -4, 13, 4, -7, -1, 1, 1, 1, -15, -11, 83, 45, -220, -88, 303, 88, -220, -45, 83, 11, -15, -1, 1, 1, 1, -31, -26, 413, 293, -3141, -1896, 15261, 7866, -50187, -22122, 115410, 43488, -189036, -60753, 222621, 60753, -189036

Offset: 1

Clark Kimberling, Nov 25 2008

The zeros of U(n,x) and U(n,-x) are the zeros of S(n,x) at A147985.

			U(3) = x^2+x-1;
U(4) = x^4+x^3-3*x^2-x+1;
U(5) = x^8+x^7-7*x^6-4*x^5+13*x^4+4*x^3-7*x^2-x+1;
so that, as an array, the sequence begins with:
1 1 -1
1 1 -3 -1 1
1 1 -7 -4 13 4 -7 -1 1

Robert Israel, Table of n, a(n) for n = 1..9418
Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Cf. A147985, A147986, A147987, A147988, A147990, A147991, A147992, A147993.

U[3]:= x^2+x-1:
U[4]:= x^4+x^3-3*x^2-x+1:
for n from 5 to 10 do
  U[n]:= normal(U[n-1]*M(U[n-1]) + x*(x^2-1)*mul(U[i]*M(U[i]),i=3..n-2));
od:
seq(seq(coeff(U[m],x,j),j=degree(U[m])..0,-1),m=3..10); # Robert Israel, Jun 30 2015

U[3, x_] = x^2 + x - 1;
U[4, x_] = x^4 + x^3 - 3 x^2 - x + 1;
U[n_, x_] := U[n, x] = U[n-1, x] U[n-1, -x] + x (x^2 - 1) Product[U[k, x] U[k, -x], {k, 3, n-2}];
Table[CoefficientList[U[n, x], x] // Reverse, {n, 3, 7}] // Flatten (* Jean-François Alcover, Mar 25 2019 *)

For n>=5, U(n)=U(n,x)=U(n-1,x)*U(n-1,-x)+x*(x^2-1)*U(3,x)*U(3,-x)*U(4,x)*U(4,-x)*...*U(n-2,x)*U(n-2,-x), where U(3)=x^2+x-1, U(4)=x^4+x^3-3*x^2-x+1.

A147990 Array A147985 (Polynomial coefficients) with zeros deleted.

Original entry on oeis.org

1, 1, -1, 1, -3, 1, 1, -7, 13, -7, 1, 1, -15, 83, -220, 303, -220, 83, -15, 1, 1, -31, 413, -3141, 15261, -50187, 115410, -189036, 222621, -189036, 115410, -50187, 15261, -3141, 413, -31, 1, 1, -63, 1839, -33150, 414861, -3841195, 27378213, -154299168

Offset: 1

Clark Kimberling, Nov 25 2008

			s(1)=x
s(2)=S(2,y)=x-1
s(3)=S(3,y)=x^2-3*x+1
s(4)=S(4,y)=x^4-7*x^3+13*x^2-7*x+1
so that as an array A147990 begins with
1
1 -1
1 -3 1
1 -7 13 -7 1

Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4

Cf. A147985, A147986, A147987, A147988, A147989, A147991, A147992, A147993.

Let s(1)=x and for n>=2, let s(n)=s(n,x)=S(n,y), where y=x^(1/2) and S(n,x)

is as at A147985. Then A147990 gives the coefficients of the polynomials s(n).

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

Original entry on oeis.org

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

Clark Kimberling, Nov 24 2008

P(n,1)=A073833(n) for n>=1; P(n,2)=A073833(n+1) for n>=0.
P(n)=P(n-1)^2+P(n-1)*P(n-2)^2-P(n-2)^4 for n>=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.

			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

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

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

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 *)

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).

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

Original entry on oeis.org

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

Clark Kimberling, Nov 24 2008

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.

			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

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

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

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).

A154305 Coefficients of polynomials H(n,x) associated with squares of polynomials S(n,x).

Original entry on oeis.org

1, 0, 1, 1, 0, -6, 0, 1, 1, 0, 20, 0, -26, 0, 20, 0, 1, 1, 0, -88, 0, 92, 0, -872, 0, 1990, 0, -872, 0, 92, 0, -88, 0, 1, 1, 0, 336, 0, -3336, 0, 6961, 0, -77796, 0, -647088, 0, 2618568, 0, -3600784, 0, 3346502, 0, -3600784, 0, 2618568, 0, -647088, 0, -77796, 0

Offset: 1

Clark Kimberling, Jan 06 2009

Define S(1)=S(1,x)=x and T(1)=T(1,x)=1; for n>=1, define S(n+1)=[S(n)]^2-[T(n)]^2 and T(n+1)=c*S(n)*T(n). The sole value of c for which S(n) is the square of a polynomial for all n>=3 is 2i, and [H(n,x)]^2 = S(n,x).

			H(3,x)=x^2+1 and S(3,x)=(x^2+1)^2.
H(4,x)=x^4-6*x^2+1
H(6,x)=x^8+20*x^6-26*x^4+20*x^2+1.
First three rows:
1 0 1
1 0 -6 0 1
1 0 20 0 -26 0 20 0 1.

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

Cf. A147985, A147986.

H(3,x)=x^2+1 and H(n+1,x)=[(2*i*x)^p]*H(n,i/(2*x)-ix/2) for n>=3, where p=2^n-2 and i=sqrt(-1).

H(n,x)=2*H(n-2,x)^4-H(n-1,x)^2. [Clark Kimberling, Mar 19 2009]

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A147985 Coefficients of numerator polynomials S(n,x) associated with reciprocation.

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A147989 Coefficients of factor polynomials U(n,x) associated with reciprocation.

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A147990 Array A147985 (Polynomial coefficients) with zeros deleted.

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

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

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A154305 Coefficients of polynomials H(n,x) associated with squares of polynomials S(n,x).

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