%I #17 Feb 04 2019 07:26:34

%S 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,

%T -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,

%U -1,3,-2,-1,4,-2,-2,3,-3,-2,5,-3,-3,5,-2,-2,6,-2,-2,5

%N 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)).

%C 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.

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

%H George E. Andrews, <a href="http://dx.doi.org/10.1006/jsco.1995.1061">On a Conjecture of Peter Borwein</a>, Journal of Symbolic Computation, Volume 20, Issues 5-6, November 1995, Pages 487-501.

%H Jiyou Li, <a href="http://arxiv.org/abs/1512.01191">A note on the Borwein conjecture</a>, arXiv:1512.01191 [math.CO], 2015.

%e For n=0, the polynomial is 1 - x - x^2 + x^3.

%e The first two rows are:

%e 1, -1, -1, 1;

%e 1, -1, -1, 1, -1, 0, 2, 0, -1, 1, -1, -1, 1.

%t 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 *)

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

%K sign,tabf

%O 0,11

%A _Michel Marcus_, Dec 04 2015