A124645 Triangle T(n,k), 0<=k<=n, read by rows given by [1,-1,0,0,0,0,0,...] DELTA [ -1,2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 .

1, 1, -1, 0, 1, -1, 0, 1, -2, 1, 0, 0, 1, -2, 1, 0, 0, 1, -3, 3, -1, 0, 0, 0, 1, -3, 3, -1, 0, 0, 0, 1, -4, 6, -4, 1, 0, 0, 0, 0, 1, -4, 6, -4, 1, 0, 0, 0, 0, 1, -5, 10, -10, 5, -1, 0, 0, 0, 0, 0, 1, -5, 10, -10, 5, -1, 0, 0, 0, 0, 0, 1, -6, 15, -20, 15, -6, 1, 0, 0, 0, 0, 0, 0, 1, -6, 15, -20, 15, -6, 1

Offset: 0

Author

Philippe Deléham, Jun 13 2007, Aug 22 2007

Keywords

• sign
• tabl

Comments

Matrix inverse of A108299.

Examples

			Triangle begins:
  1;
  1, -1;
  0,  1, -1;
  0,  1, -2,  1;
  0,  0,  1, -2,  1;
  0,  0,  1, -3,  3, -1;
  0,  0,  0,  1, -3,  3, -1;
  0,  0,  0,  1, -4,  6, -4,   1;
  0,  0,  0,  0,  1, -4,  6,  -4,   1;
  0,  0,  0,  0,  1, -5, 10, -10,   5, -1;
  0,  0,  0,  0,  0,  1, -5,  10, -10,  5, -1;
  0,  0,  0,  0,  0,  1, -6,  15, -20, 15, -6, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Programs

Magma

[(-1)^(k+Floor(n/2))*Binomial(Floor((n+1)/2), k-Floor(n/2)): k in [0..n], n in [0..12]]; // G. C. Greubel, May 01 2021

Mathematica

Table[(-1)^(Floor[n/2]+k)*Binomial[Floor[(n+1)/2], k-Floor[n/2]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 01 2021 *)

Sage

flatten([[(-1)^(k+(n//2))*binomial(((n+1)//2), k-(n//2)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 01 2021

Formula

Row n has g.f.: x^[n/2]*(1-x)^(n-[n/2]).

G.f.: (1-x*y+x)/(1-x^2*y+x^2*y^2). - R. J. Mathar, Aug 11 2015

T(n, k) = (-1)^(k + floor(n/2)) * binomial(floor((n+1)/2), k - floor(n/2)). - G. C. Greubel, May 01 2021