%I #24 May 15 2016 16:51:50
%S -1,2,1,-2,0,1,2,-3,-2,1,-2,8,0,-4,1,2,-15,10,7,-6,1,-2,24,-35,0,18,
%T -8,1,2,-35,84,-42,-30,33,-10,1,-2,48,-168,168,0,-88,52,-12,1,2,-63,
%U 300,-462,198,143,-182,75,-14,1,-2,80,-495,1056,-858,0,455,-320,102,-16,1
%N Triangle read by rows: T(n,k) = (-1)^(n-k)*(C(k+n-1,n-k)-2*C(k+n-1,n-k-1)) for n>=0 and 0<=k<=n.
%C Previous name was: "Expansion of certain polynomials; see formula."
%F B(x, 0) = -1, B(x, 2) = x^2 + 2, B(x, 3) = x^3 + x, B(x, 4) = x^4 - 2, and B(x, n) = x*B(x, n - 1) - B(x, n - 2) for n>=2, expand B(sqrt(x), 2*n).
%e {-1},
%e {2, 1},
%e {-2, 0, 1},
%e {2, -3, -2, 1},
%e {-2, 8, 0, -4, 1},
%e {2, -15, 10, 7, -6, 1},
%e {-2, 24, -35, 0, 18, -8, 1},
%e {2, -35, 84, -42, -30, 33, -10, 1},
%e {-2, 48, -168, 168,0, -88, 52, -12, 1},
%e {2, -63, 300, -462, 198, 143, -182, 75, -14,1},
%e {-2, 80, -495, 1056, -858, 0, 455, -320, 102, -16, 1}
%p T := (n,k) -> (-1)^(n-k)*(binomial(k+n-1,n-k)-2*binomial(k+n-1,n-k-1)):
%p seq(seq(T(n,k), k=0..n), n=0..10); # _Peter Luschny_, May 15 2016
%t B[x, 0] = -1; B[x, 1] = x; B[x, 2] = 2 + x^2; B[x, 4] = -2 + x^4; B[ x, 3] = x + x^3; B[x_, n_] := B[x, n] = x*B[x, n - 1] - B[x, n - 2]; a = Table[CoefficientList[B[x, n] /. x -> Sqrt[y], y], {n, 0, 20, 2}]; Flatten[a]
%Y Cf. A135929, A138034.
%K sign,tabl
%O 1,2
%A _Roger L. Bagula_, Mar 14 2008
%E Edited by _N. J. A. Sloane_, Jan 05 2009
%E Edited by _Joerg Arndt_, Nov 15 2014
%E New name and changed a(1) to -1 by _Peter Luschny_, May 15 2016