A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

1, 1, 0, 1, 2, -1, 1, 4, 3, 0, 1, 6, 15, 4, 1, 1, 8, 35, 56, 5, 0, 1, 10, 63, 204, 209, 6, -1, 1, 12, 99, 496, 1189, 780, 7, 0, 1, 14, 143, 980, 3905, 6930, 2911, 8, 1, 1, 16, 195, 1704, 9701, 30744, 40391, 10864, 9, 0, 1, 18, 255, 2716, 20305, 96030, 242047, 235416, 40545, 10, -1

Offset

0,5

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Wikipedia, Chebyshev polynomials.

Index entries for sequences related to Chebyshev polynomials.

Formula

T(0,k) = 1, T(1,k) = 2 * k and T(n,k) = 2 * k * T(n-1,k) - T(n-2,k) for n > 1.

T(n, k) = Sum_{j=0..n} (2*k-2)^j * binomial(n+1+j,2*j+1). - Seiichi Manyama, Mar 03 2021

Example

Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 2,    4,     6,      8,     10,      12, ...
  -1, 3,   15,    35,     63,     99,     143, ...
   0, 4,   56,   204,    496,    980,    1704, ...
   1, 5,  209,  1189,   3905,   9701,   20305, ...
   0, 6,  780,  6930,  30744,  96030,  241956, ...
  -1, 7, 2911, 40391, 242047, 950599, 2883167, ...

Prog

(PARI) T(n, k)  = polchebyshev(n, 2, k);
matrix(7, 7, n, k, T(n-1, k-1)) \\ Michel Marcus, Jan 07 2019
(PARI) T(n, k) = sum(j=0, n, (2*k-2)^j*binomial(n+1+j, 2*j+1)); \\ Seiichi Manyama, Mar 03 2021

Crossrefs

Mirror of A228161.

Columns 0-19 give A056594, A000027(n+1), A001353(n+1), A001109(n+1), A001090(n+1), A004189(n+1), A004191, A007655(n+2), A077412, A049660(n+1), A075843(n+1), A077421, A077423, A097309, A097311, A097313, A029548, A029547, A144128(n+1), A078987.

Rows 0-10 give A000012, A005843, A000466, A144138, A144139, A242850, A242851, A242852, A242853, A242854, A243130.

Main diagonal gives A323118.

Cf. A179943, A322836 (Chebyshev polynomial of the first kind).

Sequence in context: A157143 A112096 A217874 * A229118 A320796 A026725

Adjacent sequences: A323179 A323180 A323181 * A323183 A323184 A323185

Keyword

sign,tabl

Author

Seiichi Manyama, Jan 06 2019

Status

approved