A100551 Coefficient list of ChebyshevU(n, 1-x).

1, 2, -2, 3, -8, 4, 4, -20, 24, -8, 5, -40, 84, -64, 16, 6, -70, 224, -288, 160, -32, 7, -112, 504, -960, 880, -384, 64, 8, -168, 1008, -2640, 3520, -2496, 896, -128, 9, -240, 1848, -6336, 11440, -11648, 6720, -2048, 256, 10, -330, 3168, -13728, 32032, -43680, 35840, -17408, 4608, -512

Offset: 0

Author

Wouter Meeussen, Nov 27 2004

Keywords

• easy
• sign
• tabl

Examples

			Triangle begins as:
  1;
  2,   -2;
  3,   -8,    4;
  4,  -20,   24,    -8;
  5,  -40,   84,   -64,    16;
  6,  -70,  224,  -288,   160,    -32;
  7, -112,  504,  -960,   880,   -384,   64;
  8, -168, 1008, -2640,  3520,  -2496,  896,  -128;
  9, -240, 1848, -6336, 11440, -11648, 6720, -2048, 256;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A001353, A001787, A053117, A056594, A122803.

Programs

Magma

[Binomial(n+k+1, n-k)*(-2)^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2023

Mathematica

Table[CoefficientList[ChebyshevU[n, 1-x], x], {n, 0, 12}]

PARI

row(n) = Vecrev(polchebyshev(n, 2, 1-x)); \\ Michel Marcus, Apr 27 2020

SageMath

def A100551(n,k): return binomial(n+k+1, n-k)*(-2)^k
flatten([[A100551(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 27 2023

Formula

G.f.: ChebyshevU(n, 1-x).

From G. C. Greubel, Mar 27 2023: (Start)

T(n, k) = binomial(n+k+1, n-k)*(-2)^k.

T(n, n) = A122803(n).

T(n, n-1) = 2*(-1)^(n-1)*A001787(n), n >= 1.

Sum_{k=0..n} T(n, k) = A056594(n).

Sum_{k=0..n} (-1)^k*T(n, k) = A001353(n+1). (End)

Extensions

Keyword tabl from Michel Marcus, Apr 27 2020