login
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.
4
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.
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
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Oct 04 2020
STATUS
approved