A137350 A triangular Sequence of coefficients of a three deep polynomial recursion based on a Chebyshev kind and a Padovan recursion: Chebyshev; p(x,n)=x*p(x,n-1)-p(x,n-2); Padovan: a(n)=a(n-2)+a(n-3); Q(x, n) = x*Q(x, n - 2) - Q(x, n - 3).

1, -1, 1, 0, 1, -1, -1, 1, 1, -1, 1, 0, -2, -1, 1, 1, 2, -2, 1, -1, 1, -3, -1, 1, 0, 3, 3, -3, 1, -1, -3, 3, -4, -1, 1, 1, -1, 6, 4, -4, 1

Offset: 1

Author

Roger L. Bagula, Apr 08 2008

Keywords

• uned
• tabl
• sign

Comments

Row sums are:

{1, 0, 1, -1, 1, -2, 2, -3, 4, -5, 7}

In differential equation terms this is equivalent to ( in Mathematica notation):

D[y[x],{x,3}]=x*D[y[x],{x,1}]-y[x];

Two simple possible HypergeometricPFQ based results are:

DSolve[{D[y[x], {x, 3}] == x*D[y[x], {x, 1}] - y[x], y[0] == 1}, y, x];

DSolve[{D[y[x], {x, 3}] == x*D[y[x], {x, 1}] - y[x], y[0] == 0}, y, x].

Examples

			{1},
{-1, 1},
{0, 1},
{-1, -1, 1},
{1, -1, 1},
{0, -2, -1, 1},
{1, 2, -2, 1},
{-1, 1, -3, -1, 1},
{0, 3, 3, -3, 1},
{-1, -3, 3, -4, -1, 1},
{1, -1, 6, 4, -4, 1}
		

Crossrefs

Cf. A000931, A137276.

Programs

Mathematica

Clear[Q, x] Q[x, -2] = 1 - x; Q[x, -1] = 0; Q[x, 0] = 1; Q[x_, n_] := Q[x, n] = x*Q[x, n - 2] - Q[x, n - 3]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a]

Formula

Q(x, n) = x*Q(x, n - 2) - Q(x, n - 3).