A178120 Coefficient array of orthogonal polynomials P(n,x)=(x-2n)*P(n-1,x)-(2n-3)*P(n-2,x), P(0,x)=1,P(1,x)=x-2.

1, -2, 1, 7, -6, 1, -36, 40, -12, 1, 253, -326, 131, -20, 1, -2278, 3233, -1552, 324, -30, 1, 25059, -38140, 20678, -5260, 675, -42, 1, -325768, 523456, -310560, 90754, -14380, 1252, -56, 1, 4886521, -8205244, 5223602, -1694244, 312059, -33866, 2135, -72, 1

Offset: 0

Author

Paul Barry, May 20 2010

Keywords

• sign
• easy
• tabl

Comments

Inverse is A178121. First column is A112293 signed.

Examples

			Triangle begins
1,
-2, 1,
7, -6, 1,
-36, 40, -12, 1,
253, -326, 131, -20, 1,
-2278, 3233, -1552, 324, -30, 1,
25059, -38140, 20678, -5260, 675, -42, 1,
-325768, 523456, -310560, 90754, -14380, 1252, -56, 1,
4886521, -8205244, 5223602, -1694244, 312059, -33866, 2135, -72, 1
Production matrix of inverse is
2, 1,
1, 4, 1,
0, 3, 6, 1,
0, 0, 5, 8, 1,
0, 0, 0, 7, 10, 1,
0, 0, 0, 0, 9, 12, 1,
0, 0, 0, 0, 0, 11, 14, 1,
0, 0, 0, 0, 0, 0, 13, 16, 1,
0, 0, 0, 0, 0, 0, 0, 15, 18, 1
		

Programs

Maple

A178120 := proc(n,k)
    if n = k then
        1;
    elif n = 1 and k = 0 then
        -2 ;
    elif k < 0 or k > n then
        0 ;
    else
        -2*n*procname(n-1,k)+procname(n-1,k-1)-(2*n-3)*procname(n-2,k) ;
    end if;
end proc: # R. J. Mathar, Dec 03 2014

Mathematica

P[0, _] = 1;
P[1, x_] := x - 2;
P[n_, x_] := P[n, x] = (x-2n) P[n-1, x] - (2n-3) P[n-2, x];
T[n_] := Module[{x}, CoefficientList[P[n, x], x]];
Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Aug 06 2023 *)