A137448 Triangle T(n,k) = (1-k*(k-1))*A053120(n,k), read by rows, 0<=k<=n.

1, 0, 1, -1, 0, -2, 0, -3, 0, -20, 1, 0, 8, 0, -88, 0, 5, 0, 100, 0, -304, -1, 0, -18, 0, 528, 0, -928, 0, -7, 0, -280, 0, 2128, 0, -2624, 1, 0, 32, 0, -1760, 0, 7424, 0, -7040, 0, 9, 0, 600, 0, -8208, 0, 23616, 0, -18176, -1, 0, -50, 0, 4400, 0, -32480, 0, 70400, 0, -45568

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Apr 18 2008

Keywords

• tabl
• sign

Comments

The entries may also be defined as the coefficient [x^k] of the polynomial T(n,x)-x^2* (d^2/dx^2) T(n,x) where T are the Chebyshev polynomials (A053120).

Row sums are 1, 1, -3, -23, -79, -199, -419, -783, -1343, -2159, -3299, ...

Examples

			1;
0, 1;
-1, 0, -2;
0, -3, 0, -20;
1, 0, 8, 0, -88;
0, 5, 0, 100, 0, -304;
-1, 0, -18, 0, 528, 0, -928;
0, -7, 0, -280, 0, 2128, 0, -2624;
1, 0, 32, 0, -1760, 0, 7424, 0, -7040;
0, 9, 0, 600, 0, -8208,0, 23616, 0, -18176;
-1, 0, -50, 0, 4400, 0, -32480, 0, 70400, 0, -45568;
		

Crossrefs

Cf. A053120, A112742.

Programs

Mathematica

Clear[p, x, a] p[x, 0] = 1; p[x, 1] = x + 1; p[x_, n_] := p[x, n] = -x^2*D[ChebyshevT[n, x], {x, 2}] + ChebyshevT[n, x]; Table[Expand[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]

Formula

Row sums: Sum_{k=0..n} T(n,k) = (3+n^2-n^4)/3 = 1-A112742(n). - R. J. Mathar, Sep 10 2013

Extensions

T(1,0) corrected by R. J. Mathar, Sep 10 2013