A136093 A generalized ( Shabat) type tree transformed Chebyshev recursion where: P(x,n)->C*P(c*x+d,n)+D: C=-1;c=-1;D=1;d=1: with substitution: x-1->x; as a triangular sequence of coefficients.

1, 0, 1, -1, -1, 1, 0, -3, -1, 1, 1, 0, -4, -1, 1, 0, 3, 1, -5, -1, 1, -1, -1, 7, 2, -6, -1, 1, 0, -5, -2, 12, 3, -7, -1, 1, 1, 0, -12, -4, 18, 4, -8, -1, 1, 0, 5, 2, -24, -7, 25, 5, -9, -1, 1, -1, -1, 17, 6, -42, -11, 33, 6, -10, -1, 1

Offset: 1

Author

Roger L. Bagula, Mar 15 2008

Keywords

• uned
• tabl
• sign

Comments

Row sum is a repeating sequence:

{1, 1, -1, -3, -3, -1, 1, 1, -1, -3, -3}

Examples

			{1},
{0, 1},
{-1, -1, 1},
{0, -3, -1, 1},
{1, 0, -4, -1, 1},
{0, 3, 1, -5, -1, 1},
{-1, -1, 7, 2, -6, -1, 1},
{0, -5, -2, 12, 3, -7, -1, 1},
{1, 0, -12, -4, 18, 4, -8, -1, 1},
{0, 5, 2, -24, -7,25, 5, -9, -1, 1},
{-1, -1, 17, 6, -42, -11, 33, 6, -10, -1, 1}
		

References

• http : // logic.pdmi.ras.ru/~yumat/personaljournal/chebyshev/chebysh.htm Quote:"It is easy to see that if P is a generalized Chebyshev Polynomial, then so is polynomial CP(cz + d) + D, moreover, it represents the same tree (of course, provided that both C and c are different from zero)."

Programs

Mathematica

Clear[B, a] B[x, 0] = 1; B[x, 1] = x; B[x_, n_] := B[x, n] = -x + x*B[x, n - 1] - B[x, n - 2]; Table[ExpandAll[B[x, n]], {n, 0, 10}]; a = Table[CoefficientList[B[x, n], x], {n, 0, 10}]; Flatten[a]

Formula

p(x,0)=1;p(x,1)=x; p(x,n)=-x+x*p(x,n-1)-p(x,n-2)