A342133 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).

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, 12, 95, 448, 981, 560, 7, 0, 1, 14, 138, 900, 3344, 5346, 1912, 8, 0, 1, 16, 189, 1584, 8525, 24960, 29133, 6528, 9, 0, 1, 18, 248, 2548, 18180, 80750, 186304, 158760, 22288, 10, 0

Offset: 0

Author

Seiichi Manyama, Mar 01 2021

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1, 1,   1,    1,     1,     1, ...
  0, 2,   4,    6,     8,    10, ...
  0, 3,  14,   33,    60,    95, ...
  0, 4,  48,  180,   448,   900, ...
  0, 5, 164,  981,  3344,  8525, ...
  0, 6, 560, 5346, 24960, 80750, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

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

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

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

Cf. A342120, A342129, A342134.

Programs

Maple

T:= (n, k)-> (<<0|1>, <-k|2*k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 01 2021

Mathematica

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 *)

PARI

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

PARI

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

PARI

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

Formula

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.

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).

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