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A117944 Triangle related to powers of 3 partitions of n.

%I A117944 #10 Nov 12 2021 00:53:01
%S A117944 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,
%T A117944 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,
%U A117944 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
%N A117944 Triangle related to powers of 3 partitions of n.
%C A117944 Inverse is A117945.
%C A117944 Row sums of inverse are A039966.
%H A117944 G. C. Greubel, <a href="/A117944/b117944.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A117944 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.
%F A117944 T(n, k) = A117939(n,k) mod 2.
%F A117944 T(n, k) = A117939^(-1)(n,k) mod 2.
%F A117944 Sum_{k=0..n} T(n, k) = A117943(n).
%e A117944 Triangle begins
%e A117944   1;
%e A117944   0, 1;
%e A117944   1, 0, 1;
%e A117944   0, 0, 0, 1;
%e A117944   0, 0, 0, 0, 1;
%e A117944   0, 0, 0, 1, 0, 1;
%e A117944   1, 0, 0, 0, 0, 0, 1;
%e A117944   0, 1, 0, 0, 0, 0, 0, 1;
%e A117944   1, 0, 1, 0, 0, 0, 1, 0, 1;
%e A117944   0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A117944   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A117944   0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1;
%t A117944 T[n_, k_]:= Mod[Sum[JacobiSymbol[Binomial[n, j], 3]*JacobiSymbol[Binomial[n-j, k], 3], {j,0,n}], 2];
%t A117944 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 29 2021 *)
%o A117944 (Sage)
%o A117944 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
%o A117944 flatten([[A117944(n, k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Oct 29 2021
%Y A117944 Cf. A039966, A117939, A117943, A117945.
%K A117944 easy,nonn,tabl
%O A117944 0,1
%A A117944 _Paul Barry_, Apr 05 2006