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A147989 Coefficients of factor polynomials U(n,x) associated with reciprocation. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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

LINKS

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.

FORMULA

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.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Mar 25 2019 *)

CROSSREFS

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

Sequence in context: A208615 A058663 A124371 * A119329 A054724 A061494

Adjacent sequences:  A147986 A147987 A147988 * A147990 A147991 A147992

KEYWORD

sign,tabf,look

AUTHOR

Clark Kimberling, Nov 25 2008

STATUS

approved

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Last modified July 18 19:00 EDT 2019. Contains 325144 sequences. (Running on oeis4.)