A117945 Triangle related to powers of 3 partitions of n.

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

Offset: 0

Author

Paul Barry, Apr 05 2006

Keywords

• sign
• tabl

Comments

Row sums are A039966.

Inverse of A117944.

Examples

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

Links

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

Crossrefs

Cf. A039966, A117944.

Programs

Mathematica

M[n_, k_]:= M[n, k] = If[k>n, 0, Mod[Sum[JacobiSymbol[Binomial[n, j], 3]*JacobiSymbol[Binomial[n-j, k], 3], {j,0,n}], 2], 0];
m:= m= With[{q=60}, 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 29 2021 *)