%I #47 Mar 03 2021 12:16:46

%S 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,

%T 12,99,496,1189,780,7,0,1,14,143,980,3905,6930,2911,8,1,1,16,195,1704,

%U 9701,30744,40391,10864,9,0,1,18,255,2716,20305,96030,242047,235416,40545,10,-1

%N 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.

%H Seiichi Manyama, <a href="/A323182/b323182.txt">Antidiagonals n = 0..139, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F 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.

%F T(n, k) = Sum_{j=0..n} (2*k-2)^j * binomial(n+1+j,2*j+1). - _Seiichi Manyama_, Mar 03 2021

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 2, 4, 6, 8, 10, 12, ...

%e -1, 3, 15, 35, 63, 99, 143, ...

%e 0, 4, 56, 204, 496, 980, 1704, ...

%e 1, 5, 209, 1189, 3905, 9701, 20305, ...

%e 0, 6, 780, 6930, 30744, 96030, 241956, ...

%e -1, 7, 2911, 40391, 242047, 950599, 2883167, ...

%o (PARI) T(n,k) = polchebyshev(n, 2, k);

%o matrix(7, 7, n, k, T(n-1,k-1)) \\ _Michel Marcus_, Jan 07 2019

%o (PARI) T(n, k) = sum(j=0, n, (2*k-2)^j*binomial(n+1+j, 2*j+1)); \\ _Seiichi Manyama_, Mar 03 2021

%Y Mirror of A228161.

%Y 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.

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

%Y Main diagonal gives A323118.

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

%K sign,tabl

%O 0,5

%A _Seiichi Manyama_, Jan 06 2019