A144388 Triangle T(n,k) = binomial(n, k) + ((-1)^(n + k))*n*binomial(n - 1, k), T(0,0) = 1, read by rows, 0 <= k <= n.
1, 0, 1, 3, 0, 1, -2, 9, 0, 1, 5, -8, 18, 0, 1, -4, 25, -20, 30, 0, 1, 7, -24, 75, -40, 45, 0, 1, -6, 49, -84, 175, -70, 63, 0, 1, 9, -48, 196, -224, 350, -112, 84, 0, 1, -8, 81, -216, 588, -504, 630, -168, 108, 0, 1, 11, -80, 405, -720, 1470, -1008, 1050, -240, 135, 0, 1
Offset: 0
Examples
Triangle begins: 1; 0, 1; 3, 0, 1; -2, 9, 0, 1; 5, -8, 18, 0, 1; -4, 25, -20, 30, 0, 1; 7, -24, 75, -40, 45, 0, 1; -6, 49, -84, 175, -70, 63, 0, 1; 9, -48, 196, -224, 350, -112, 84, 0, 1; -8, 81, -216, 588, -504, 630, -168, 108, 0, 1; 11, -80, 405, -720, 1470, -1008, 1050, -240, 135, 0, 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(binomial(n, k) + ((-1)^(n + k))*n*binomial(n - 1, k),n , 0, 15, k, 0, n); /* Franck Maminirina Ramaharo, Jan 25 2019 */
Formula
T(n,k) = [x^k] ((x + 1)^n - n*(x - 1)^(n - 1)).
Sum_{k=0..n} T(n,k) = A151821(n-1), n >= 1.
Extensions
Edited and offset corrected by Franck Maminirina Ramaharo, Jan 25 2019