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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 19:04 EST 2022. Contains 358588 sequences. (Running on oeis4.)