A117906 Inverse of number triangle A117904.

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

Offset: 0

Author

Paul Barry, Apr 01 2006

Keywords

• easy
• sign
• tabl

Comments

Row sums are (1, 0, 1, 0, 0, 0, ...) with g.f. 1 + x^2.

Diagonal sums are A117907.

Examples

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

Links

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

Crossrefs

Cf. A117904, A117907.

Programs

Mathematica

M[n_, k_]:= M[n, k]= If[k>n, 0, If[Abs[JacobiSymbol[Binomial[n, 2], 3] - JacobiSymbol[Binomial[k, 2], 3]]==0, 1, 0]];
m:= m= With[{q=20}, Table[M[n, k], {n,0,q}, {k,0,q}]];
T[n_, k_]:= Inverse[m][[n+1, k+1]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 20 2021 *)

Formula

G.f.: (1 -x*(1-y) +x^2*y^2 -x^3*y -x^5*y^2)/(1-x^3*y^3).