A342134 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, 5, 0, 1, 6, 18, 12, 0, 1, 8, 39, 80, 29, 0, 1, 10, 68, 252, 356, 70, 0, 1, 12, 105, 576, 1629, 1584, 169, 0, 1, 14, 150, 1100, 4880, 10530, 7048, 408, 0, 1, 16, 203, 1872, 11525, 41344, 68067, 31360, 985, 0, 1, 18, 264, 2940, 23364, 120750, 350272, 439992, 139536, 2378, 0

Offset

0,5

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Index entries for sequences related to Chebyshev polynomials.

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

Example

Square array begins:
  1,  1,    1,     1,     1,      1, ...
  0,  2,    4,     6,     8,     10, ...
  0,  5,   18,    39,    68,    105, ...
  0, 12,   80,   252,   576,   1100, ...
  0, 29,  356,  1629,  4880,  11525, ...
  0, 70, 1584, 10530, 41344, 120750, ...

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 *)

Prog

(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)*I)^n*polchebyshev(n, 2, sqrt(k)*I));

Crossrefs

Columns 0..5 give A000007, A000129(n+1), A090017(n+1), A090018, A190510(n+1), A190955(n+1).

Rows 0..2 give A000012, A005843, A007742.

Main diagonal gives A109517(n+1).

Cf. A342120, A342129, A342133.

Sequence in context: A139435 A077909 A247126 * A349740 A327117 A359107

Adjacent sequences: A342131 A342132 A342133 * A342135 A342136 A342137

Keyword

nonn,tabl

Author

Seiichi Manyama, Mar 01 2021

Status

approved