A178107 Coefficient array for orthogonal polynomials P(n,x)=x*P(n-1,x)-(2*floor((n+2)/2)-3)*P(n-2,x), P(0,x)=1,P(1,x)=x.

1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 3, 0, -5, 0, 1, 0, 9, 0, -8, 0, 1, -15, 0, 34, 0, -13, 0, 1, 0, -60, 0, 74, 0, -18, 0, 1, 105, 0, -298, 0, 165, 0, -25, 0, 1, 0, 525, 0, -816, 0, 291, 0, -32, 0, 1, -945, 0, 3207, 0, -2301, 0, 516, 0, -41, 0, 1, 0, -5670, 0, 10551, 0, -4920, 0, 804, 0

Offset: 0

Author

Paul Barry, May 20 2010

Keywords

• sign
• tabl

Comments

Inverse is A178108. First column is signed aerated version of double factorials A001147.

Examples

			Triangle begins
1,
0, 1,
-1, 0, 1,
0, -2, 0, 1,
3, 0, -5, 0, 1,
0, 9, 0, -8, 0, 1,
-15, 0, 34, 0, -13, 0, 1,
0, -60, 0, 74, 0, -18, 0, 1,
105, 0, -298, 0, 165, 0, -25, 0, 1,
0, 525, 0, -816, 0, 291, 0, -32, 0, 1,
-945, 0, 3207, 0, -2301, 0, 516, 0, -41, 0, 1
Production matrix is
0, 1,
-1, 0, 1,
0, -1, 0, 1,
1, 0, -3, 0, 1,
0, 1, 0, -3, 0, 1,
2, 0, 1, 0, -5, 0, 1,
0, 2, 0, 1, 0, -5, 0, 1,
7, 0, 2, 0, 1, 0, -7, 0, 1,
0, 7, 0, 2, 0, 1, 0, -7, 0, 1
Production matrix of inverse is
0, 1,
1, 0, 1,
0, 1, 0, 1,
0, 0, 3, 0, 1,
0, 0, 0, 3, 0, 1,
0, 0, 0, 0, 5, 0, 1,
0, 0, 0, 0, 0, 5, 0, 1,
0, 0, 0, 0, 0, 0, 7, 0, 1,
0, 0, 0, 0, 0, 0, 0, 7, 0, 1
		

Programs

Mathematica

p[0] = 1; p[1] = x;
p[n_] := p[n] = Expand[x p[n-1] - (2 Quotient[n+2, 2] - 3) p[n-2]];
Table[CoefficientList[p[n], x], {n, 0, 6}] (* Andrey Zabolotskiy, Dec 26 2023 *)