%I #10 Jan 26 2019 05:00:23

%S -1,2,-1,1,4,-1,4,3,6,-1,3,16,6,8,-1,6,15,40,10,10,-1,5,36,45,80,15,

%T 12,-1,8,35,126,105,140,21,14,-1,7,64,140,336,210,224,28,16,-1,10,63,

%U 288,420,756,378,336,36,18,-1,9,100,315,960,1050,1512,630,480,45,20,-1

%N Triangle T(n,k) = n*binomial(n - 1, k) - (-1)^(n - k)*binomial(n, k), T(0,0) = 1, read by rows, 0 <= k <= n.

%F T(n,k) = [x^k] (n*(x + 1)^(n - 1) - (x - 1)^n).

%F Sum_{k=0..n} T(n,k) = A001787(n), n >= 1.

%e Triangle begins:

%e -1;

%e 2, -1;

%e 1, 4, -1;

%e 4, 3, 6, -1;

%e 3, 16, 6, 8, -1;

%e 6, 15, 40, 10, 10, -1;

%e 5, 36, 45, 80, 15, 12, -1;

%e 8, 35, 126, 105, 140, 21, 14, -1;

%e 7, 64, 140, 336, 210, 224, 28, 16, -1;

%e 10, 63, 288, 420, 756, 378, 336, 36, 18, -1;

%e 9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1;

%e ...

%t p[x_, n_] = -(x - 1)^n + n*(x + 1)^(n - 1);

%t Table[CoefficientList[p[x, n], x], {n, 0, 10}] // Flatten

%o (Maxima) create_list(n*binomial(n - 1, k) - (-1)^(n - k)*binomial(n, k), n , 0, 15, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 25 2019 */

%Y Cf. A001787, A007318, A130595, A144388, A216973.

%K sign,easy,tabl

%O 0,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 01 2008