%I #36 Dec 12 2023 20:22:01

%S 1,1,0,1,2,0,1,4,3,0,1,6,14,4,0,1,8,33,48,5,0,1,10,60,180,164,6,0,1,

%T 12,95,448,981,560,7,0,1,14,138,900,3344,5346,1912,8,0,1,16,189,1584,

%U 8525,24960,29133,6528,9,0,1,18,248,2548,18180,80750,186304,158760,22288,10,0

%N Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2*k*x + k*x^2).

%H Seiichi Manyama, <a href="/A342133/b342133.txt">Antidiagonals n = 0..139, flattened</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) = k*(2*T(n-1,k) - T(n-2,k)) for n > 1.

%F T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (-1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (-1/2)^(n-j) * binomial(j,n-j).

%F T(n,k) = sqrt(k)^n * U(n, sqrt(k)) where U(n, x) is a Chebyshev polynomial of the second kind.

%e Square array begins:

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

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

%e 0, 3, 14, 33, 60, 95, ...

%e 0, 4, 48, 180, 448, 900, ...

%e 0, 5, 164, 981, 3344, 8525, ...

%e 0, 6, 560, 5346, 24960, 80750, ...

%p T:= (n, k)-> (<<0|1>, <-k|2*k>>^(n+1))[1, 2]:

%p seq(seq(T(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Mar 01 2021

%t T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * (-2)^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Apr 27 2021 *)

%o (PARI) T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*(-2)^(-j)*binomial(n-j, j));

%o (PARI) T(n, k) = sum(j=0, n, (2*k)^j*(-2)^(j-n)*binomial(j, n-j));

%o (PARI) T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)));

%Y Columns 0..5 give A000007, A000027(n+1), A007070, A138395, A099156(n+1), A190987(n+1).

%Y Rows 0..2 give A000012, A005843, A033991.

%Y Main diagonal gives (-1) * A109520(n+1).

%Y Cf. A342120, A342129, A342134.

%K nonn,tabl

%O 0,5

%A _Seiichi Manyama_, Mar 01 2021