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A117944
Triangle related to powers of 3 partitions of n.
5
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.
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
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Apr 05 2006
STATUS
approved