A127793 Inverse of number triangle A(n,k) = 1/floor((n+2)/2) if k <= n <= 2k, 0 otherwise.

1, 0, 1, 0, -1, 2, 0, 1, -2, 2, 0, 0, 0, -2, 3, 0, -1, 2, 0, -3, 3, 0, 0, 0, 0, 0, -3, 4, 0, 1, -2, 2, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 0, -4, 5, 0, 0, 0, -2, 3, 0, 0, 0, -5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 6

Offset: 0

Author

Paul Barry, Jan 29 2007

Keywords

• sign
• tabl

Comments

It is conjectured that the triangle is an integer triangle. The triangle and its inverse both appear to have row sums equal to the all 1's sequence.

The triangle is equivalent to the lower semi-matrix R = e_{1,1} + Sum_{i>=2} Sum_{p>=0} ( e_{2^p i, i} ceiling(i/2) - e_{2^p (i+1), i} ceiling(i/2) ) , where e_{i,j} is the matrix unit. The conjecture above is true, deduced from the formula of the matrix. - FUNG Cheok Yin, Sep 12 2022

Examples

			Triangle begins
  1;
  0,  1;
  0, -1,  2;
  0,  1, -2,  2;
  0,  0,  0, -2,  3;
  0, -1,  2,  0, -3,  3;
  0,  0,  0,  0,  0, -3,  4;
  0,  1, -2,  2,  0,  0, -4,  4;
  0,  0,  0,  0,  0,  0,  0, -4,  5;
  0,  0,  0, -2,  3,  0,  0,  0, -5,  5;
  0,  0,  0,  0,  0,  0,  0,  0,  0, -5,  6;
  0, -1,  2,  0, -3,  3,  0,  0,  0,  0, -6,  6;
  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -6,  7;
Inverse of the triangle begins
  1;
  0,  1;
  0, 1/2, 1/2;
  0,  0,  1/2, 1/2;
  0,  0,  1/3, 1/3, 1/3;
  0,  0,   0,  1/3, 1/3, 1/3;
  0,  0,   0,  1/4, 1/4, 1/4, 1/4;
  0,  0,   0,   0,  1/4, 1/4, 1/4, 1/4;
  0,  0,   0,   0,  1/5, 1/5, 1/5, 1/5, 1/5;
  0,  0,   0,   0,   0,  1/5, 1/5, 1/5, 1/5, 1/5;
  0,  0,   0,   0,   0,  1/6, 1/6, 1/6, 1/6, 1/6, 1/6;
		

Links

• FUNG Cheok Yin, the triangle with the first 61 rows

Programs

Mathematica

rows = 11;
A[n_, k_] := If[k <= n, If[n <= 2 k, 1/Floor[(n+2)/2] , 0], 0];
T = Table[A[n, k], {n, 0, rows-1}, {k, 0, rows-1}] // Inverse;
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Stefano Spezia, Sep 30 2018 *)