%I #3 Mar 30 2012 17:34:22
%S 1,0,-1,0,-2,1,0,0,3,-1,0,0,1,-4,1,0,0,-1,-2,5,-1,0,0,1,2,3,-6,1,0,0,
%T -1,-2,-3,-4,7,-1,0,0,1,2,3,4,5,-8,1,0,0,-1,-2,-3,-4,-5,-6,9,-1,0,0,1,
%U 2,3,4,5,6,7,-10,1,0,0,-1,-2,-3,-4,-5,-6,-7,-8,11,-1
%N Integral form based on A122188: Coefficients of recursive polynomial from: p(x,n)=x*IP(x,n)=-(1+x)*B(x,n)-B'(x,n).
%C Row sum:
%C {1, -1, -1, 2, -2, 1, 1, -4, 8, -13, 19, -26}
%C Derivation:
%C The Bonacci polynomial recursion:
%C B(x, n) = -(1 + x)*B(x, n - 1) - x*B(x, n - 2);
%C Implies the differential equation:
%C B''(x, n) = -(1 + x)*B'(x, n) - x*B(x, n);
%C Integration to a constant of that differential equation gives:
%C B'(x,n)=-(1+x)*B(x,n)-x*IB(x,n)
%C which rearranged gives the output form given here.
%F Bonacci polynomial recursion: B(x, n) = -(1 + x)*B(x, n - 1) - x*B(x, n - 2); Differentiation Polynomials: B'(x, n) := D[B[x, n + 1], x]=dB[x,n+1)/dx; Output form: p(x,n)=x*IP(x,n)=-(1+x)*B(x,n)-B'(x,n)
%e {1},
%e {0, -1},
%e {0, -2, 1},
%e {0, 0, 3, -1},
%e {0, 0, 1, -4, 1},
%e {0, 0, -1, -2, 5, -1},
%e {0, 0, 1, 2, 3, -6, 1},
%e {0, 0, -1, -2, -3, -4, 7, -1},
%e {0, 0, 1, 2, 3,4, 5, -8, 1},
%e {0, 0, -1, -2, -3, -4, -5, -6, 9, -1},
%e {0, 0, 1, 2, 3, 4,5, 6, 7, -10, 1},
%e {0, 0, -1, -2, -3, -4, -5, -6, -7, -8, 11, -1}
%t Clear[B, x, n] B[x, 0] = 1; B[x, 1] = -1*x + 1; B[x_, n_] := B[x, n] = -(1 + x)*B[x, n - 1] - x*B[x, n - 2]; P[x_, n_] := D[B[x, n + 1], x]; Table[ExpandAll[ -(1 + x)*B[x, n] - P[x, n]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[ -(1 + x)* B[x, n] - P[x, n], x], {n, 0, 10}]]; Flatten[a] Table[Apply[Plus, CoefficientList[ -(1 + x)*B[x, n] - P[x, n], x]], {n, 0, 10}];
%Y Cf. A122188.
%K uned,tabl,sign
%O 1,5
%A _Roger L. Bagula_, Mar 18 2008