A117944 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

0,1

Comments

Inverse is A117945.

Row sums of inverse are A039966.

Links

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

Formula

Triangle T(n,k) = Sum_{j=0..n} L(C(n,j)/3)*L(C(n-j,k)/3) mod 2, where L(j/p) is the Legendre symbol of j and p.

T(n, k) = A117939(n,k) mod 2.

T(n, k) = A117939^(-1)(n,k) mod 2.

Sum_{k=0..n} T(n, k) = A117943(n).

Example

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;

Mathematica

T[n_, k_]:= Mod[Sum[JacobiSymbol[Binomial[n, j], 3]*JacobiSymbol[Binomial[n-j, k], 3], {j, 0, n}], 2];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 29 2021 *)

Prog

(Sage)
def A117944(n, k): return ( sum(jacobi_symbol(binomial(n, j), 3)*jacobi_symbol(binomial(n-j, k), 3) for j in (0..n)) )%2
flatten([[A117944(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Oct 29 2021

Crossrefs

Cf. A039966, A117939, A117943, A117945.

Sequence in context: A357518 A093317 A127253 * A117945 A128937 A288004

Adjacent sequences: A117941 A117942 A117943 * A117945 A117946 A117947

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 05 2006

Status

approved