A115718 Inverse of number triangle A115717; a divide-and-conquer related triangle.

1, 0, 1, -3, 1, 1, 0, 0, 0, 1, -3, -3, 1, 1, 1, 0, 0, 0, 0, 0, 1, -3, -3, -3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, -3, -3, -3, -3, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -3, -3, -3, -3, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -3, -3, -3, -3, -3, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Paul Barry, Jan 29 2006

Keywords

• easy
• sign
• tabl

Comments

Product of A115713 and (1/(1-x), x).

Row sums are 1,1,-1,1,-3,1,-5,1,-7,1, ... with g.f. (1+x-3*x^2-x^3)/(1-x^2)^2.

Row sums of inverse are A115716.

Examples

			Triangle begins
   1;
   0,  1;
  -3,  1,  1;
   0,  0,  0,  1;
  -3, -3,  1,  1,  1;
   0,  0,  0,  0,  0,  1;
  -3, -3, -3,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3,  1,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1,  1,  1;
		

Links

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

Crossrefs

Cf. A115713, A115716, A115717, A237420.

Programs

Mathematica

T[n_, k_]:= If[OddQ[n], If[kG. C. Greubel, Nov 29 2021 *)

Sage

def A115718(n,k):
    if (n%2==0): return 0 if (kA115718(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 29 2021

Formula

From G. C. Greubel, Nov 29 2021: (Start)

T(2*n, k) = -3 if (k < n/2) otherwise 1.

T(2*n+1, k) = 0 if (k < n) otherwise 1.

Sum_{k=0..n} T(n, k) = (1/2)*(2 + (1 + (-1)^n)*n) = 1 + A237420(n). (End)