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

0,8

Comments

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

Links

Table of n, a(n) for n=0..74.

Example

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

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 *)

Crossrefs

Sequence in context: A263730 A331533 A089994 * A272472 A355536 A100260

Adjacent sequences: A178104 A178105 A178106 * A178108 A178109 A178110

Keyword

sign,tabl

Author

Paul Barry, May 20 2010

Status

approved