%I #3 Mar 30 2012 17:34:26
%S 1,0,1,0,1,1,0,2,-1,0,2,0,-1,0,2,1,-1,-1,0,2,1,-2,1,0,3,-1,-2,0,1,0,3,
%T -1,-2,-1,1,1,0,3,0,-3,-1,2,-1,0,3,0,-4,1,2,0,-1
%N Triangular sequence of coefficients from a polynomial recursion: p(x,n)=p(x,Floor[(n-1)/2])-x^2*p(x,n-3)+x.
%C Row sums are: {1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 1, ...}
%F p(x,n)=p(x,Floor[(n-1)/2])-x^2*p(x,n-3)+x; out_n,m=Coefficient(p(x,n)).
%e {1},
%e {0, 1},
%e {0, 1, 1},
%e {0, 2, -1},
%e {0, 2, 0, -1},
%e {0, 2, 1, -1, -1},
%e {0, 2, 1, -2, 1},
%e {0, 3, -1, -2, 0, 1},
%e {0, 3, -1, -2, -1,1, 1},
%e {0, 3, 0, -3, -1, 2, -1},
%e {0, 3, 0, -4, 1, 2, 0, -1}
%t Clear[p, x]; p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = x; p[x, 2] = x^2 + x; p[x_, n_] := p[x, n] = p[x, Floor[(n - 1)/2]] - x^2*p[x, n - 3] + x; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[p[x, n], x]], {n, 0, 10}];
%K tabl,sign
%O 1,8
%A _Roger L. Bagula_, Apr 27 2008