A265196 Irregular triangle read by rows, where T(n, k) is the coefficient of degree k of the polynomial Product_{j=0..n} (1-x^(3*j+1))*(1-x^(3*j+2)).

1, -1, -1, 1, 1, -1, -1, 1, -1, 0, 2, 0, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 0, 2, -1, -1, 3, -1, -1, 2, -2, -2, 2, -1, -1, 3, -1, -1, 2, 0, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 0, 2, -1, -1, 3, -2, -1, 4, -2, -2, 3, -3, -2, 5, -3, -3, 5, -2, -2, 6, -2, -2, 5

Offset

0,11

Comments

Borwein conjectures that T(n,k) >= 0 when k is a multiple of 3, and T(n,k) <= 0 is not a multiple of 3.

The length of the 0th row is 4 and, for n > 0, the length of the n-th row is 3*n^2+1.

Links

Table of n, a(n) for n=0..72.

George E. Andrews, On a Conjecture of Peter Borwein, Journal of Symbolic Computation, Volume 20, Issues 5-6, November 1995, Pages 487-501.

Jiyou Li, A note on the Borwein conjecture, arXiv:1512.01191 [math.CO], 2015.

Example

For n=0, the polynomial is 1 - x - x^2 + x^3.
The first two rows are:
1, -1, -1, 1;
1, -1, -1, 1, -1, 0, 2, 0, -1, 1, -1, -1, 1.

Mathematica

row[n_] := CoefficientList[Product[(1-x^(3j+1))(1-x^(3j+2)), {j, 0, n}], x]; Table[row[n], {n, 0, 3}] // Flatten (* Jean-François Alcover, Sep 27 2018 *)

Prog

(PARI) row(n) = Vec(prod(j=0, n, (1-x^(3*j+1))*(1-x^(3*j+2))));

Crossrefs

Sequence in context: A317992 A228085 A154782 * A171157 A194301 A194341

Adjacent sequences: A265193 A265194 A265195 * A265197 A265198 A265199

Keyword

sign,tabf

Author

Michel Marcus, Dec 04 2015

Status

approved