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

Author

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

Keywords

• sign
• easy
• tabl

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;
  ...
		

Crossrefs

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

Programs

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.