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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
OFFSET
0,5
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.
Main diagonal gives A323118.
Cf. A179943, A322836 (Chebyshev polynomial of the first kind).
Sequence in context: A157143 A112096 A217874 * A229118 A320796 A026725
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Jan 06 2019
STATUS
approved