A337966 Triangle read by rows, coefficients of polynomials over {-1, 0, 1}. Also a triangle-to-triangle transformation U -> T(U) applied to the triangle U(n, k) = 1.

1, 1, 0, -1, -1, 1, 0, -1, 0, 1, -1, 1, 1, -1, -1, -1, 0, 1, 0, -1, 0, 1, 1, -1, -1, 1, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1

Offset

0

Comments

The triangle can also be seen as a generalization of A118828.

Links

Table of n, a(n) for n=0..77.

Formula

Let polynomials P(n, z) be defined by:

t(n, x) = Sum_{k=0..n} z^k*x^(n-k).

s(n, x) = x^n*t(n, -x)/(1 - (-x))^(n+1).

S(n, x) = x*(s(n, x) - s(n, -x)). Let i denote the imaginary unit.

Then P(n, z) = (-2)^floor(n/2)*S(n, i) and T(n, k) = [z^k] P(n, z).

Example

Triangle starts:
[0]  1
[1]  1,  0
[2] -1, -1,  1
[3]  0, -1,  0,  1
[4] -1,  1,  1, -1, -1
[5] -1,  0,  1,  0, -1,  0
[6]  1,  1, -1, -1,  1,  1, -1
[7]  0,  1,  0, -1,  0,  1,  0, -1
[8]  1, -1, -1,  1,  1, -1, -1,  1, 1
[9]  1,  0, -1,  0,  1,  0, -1,  0, 1, 0

Maple

A337966 := proc(n, k) [1, 1, -1, 0, -1, -1, 1, 0][irem(n + 2*k, 8) + 1] end:
for n from 0 to 9 do lprint(seq(A337966(n, k), k=0..n)) od;

Crossrefs

Cf. A118828 (diagonal, column 0 and row sum, with some shifts).

Cf. A337967 (shows an interpretation as a transform).

Sequence in context: A267778 A285384 A165728 * A010890 A011633 A015254

Adjacent sequences: A337963 A337964 A337965 * A337967 A337968 A337969

Keyword

sign,tabl

Author

Peter Luschny, Oct 04 2020

Status

approved