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-François Alcover, Mar 25 2019 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Nov 25 2008
STATUS
approved