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

%I A147989 #17 Mar 25 2019 10:07:17
%S A147989 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,
%T A147989 303,88,-220,-45,83,11,-15,-1,1,1,1,-31,-26,413,293,-3141,-1896,15261,
%U A147989 7866,-50187,-22122,115410,43488,-189036,-60753,222621,60753,-189036
%N A147989 Coefficients of factor polynomials U(n,x) associated with reciprocation.
%C A147989 The zeros of U(n,x) and U(n,-x) are the zeros of S(n,x) at A147985.
%H A147989 Robert Israel, <a href="/A147989/b147989.txt">Table of n, a(n) for n = 1..9418</a>
%H A147989 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Kimberling/kimberling56.html">Polynomials associated with reciprocation</a>, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.
%F A147989 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.
%e A147989 U(3) = x^2+x-1;
%e A147989 U(4) = x^4+x^3-3*x^2-x+1;
%e A147989 U(5) = x^8+x^7-7*x^6-4*x^5+13*x^4+4*x^3-7*x^2-x+1;
%e A147989 so that, as an array, the sequence begins with:
%e A147989 1 1 -1
%e A147989 1 1 -3 -1 1
%e A147989 1 1 -7 -4 13 4 -7 -1 1
%p A147989 U[3]:= x^2+x-1:
%p A147989 U[4]:= x^4+x^3-3*x^2-x+1:
%p A147989 for n from 5 to 10 do
%p A147989   U[n]:= normal(U[n-1]*M(U[n-1]) + x*(x^2-1)*mul(U[i]*M(U[i]),i=3..n-2));
%p A147989 od:
%p A147989 seq(seq(coeff(U[m],x,j),j=degree(U[m])..0,-1),m=3..10); # _Robert Israel_, Jun 30 2015
%t A147989 U[3, x_] = x^2 + x - 1;
%t A147989 U[4, x_] = x^4 + x^3 - 3 x^2 - x + 1;
%t A147989 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}];
%t A147989 Table[CoefficientList[U[n, x], x] // Reverse, {n, 3, 7}] // Flatten (* _Jean-François Alcover_, Mar 25 2019 *)
%Y A147989 Cf. A147985, A147986, A147987, A147988, A147990, A147991, A147992, A147993.
%K A147989 sign,tabf,look
%O A147989 1,6
%A A147989 _Clark Kimberling_, Nov 25 2008