A117939 Triangle related to powers of 3 partitions of n.

1, 2, 1, 1, -2, 1, 2, 0, 0, 1, 4, 2, 0, 2, 1, 2, -4, 2, 1, -2, 1, 1, 0, 0, -2, 0, 0, 1, 2, 1, 0, -4, -2, 0, 2, 1, 1, -2, 1, -2, 4, -2, 1, -2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, -4, 2, 0, 0, 0, 0, 0, 0, 1, -2, 1, 4, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 8, 4, 0, 4, 2, 0, 0, 0, 0, 4, 2, 0, 2, 1

Offset: 0

Author

Paul Barry, Apr 05 2006

Keywords

• easy
• sign
• tabl

Examples

			Triangle begins
  1;
  2,  1;
  1, -2, 1;
  2,  0, 0,  1;
  4,  2, 0,  2,  1;
  2, -4, 2,  1, -2,  1;
  1,  0, 0, -2,  0,  0, 1;
  2,  1, 0, -4, -2,  0, 2,  1;
  1, -2, 1, -2,  4, -2, 1, -2, 1;
		

Links

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

Crossrefs

Cf. A059151, A117940, A117944, A117946.

Cf. A120854 (matrix log), A117941 (inverse), A117947 (matrix square-root).

Programs

Mathematica

T[n_, k_]:= Sum[JacobiSymbol[Binomial[n, j], 3]*JacobiSymbol[Binomial[n-j, k], 3], {j, 0, n}]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 29 2021 *)

PARI

T(n,k)=(matrix(n+1,n+1,r,c,(binomial(r-1,c-1)+1)%3-1)^2)[n+1,k+1] \\ Paul D. Hanna, Jul 08 2006

Sage

def A117939(n, k): return sum(jacobi_symbol(binomial(n, j), 3)*jacobi_symbol(binomial(n-j, k), 3) for j in (0..n))
flatten([[A117939(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Oct 29 2021

Formula

Triangle T(n,k) = Sum_{j=0..n} L(C(n,j)/3)*L(C(n-j,k)/3) where L(j/p) is the Legendre symbol of j and p.

T(n, k) mod 2 = A117944(n,k).

T(n, 0) = A059151(n).

T(n, 1) = A117946(n).

Sum_{k=0..n} T(n, k) = A117940(n).

Matrix square of triangle A117947. Matrix log is the integer triangle A120854. - Paul D. Hanna, Jul 08 2006