A144389 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.
-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, 12, -1, 8, 35, 126, 105, 140, 21, 14, -1, 7, 64, 140, 336, 210, 224, 28, 16, -1, 10, 63, 288, 420, 756, 378, 336, 36, 18, -1, 9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1
Offset: 0
Examples
Triangle begins: -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, 12, -1; 8, 35, 126, 105, 140, 21, 14, -1; 7, 64, 140, 336, 210, 224, 28, 16, -1; 10, 63, 288, 420, 756, 378, 336, 36, 18, -1; 9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1; ...
Programs
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Mathematica
p[x_, n_] = -(x - 1)^n + n*(x + 1)^(n - 1); Table[CoefficientList[p[x, n], x], {n, 0, 10}] // Flatten -
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 */
Formula
T(n,k) = [x^k] (n*(x + 1)^(n - 1) - (x - 1)^n).
Sum_{k=0..n} T(n,k) = A001787(n), n >= 1.