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

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

Author

Clark Kimberling, Nov 25 2008

Keywords

• sign
• tabf
• look

Comments

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

Examples

			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
		

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.

Crossrefs

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

Programs

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-François Alcover, Mar 25 2019 *)

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.