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A117939 Triangle related to powers of 3 partitions of n.

%I A117939 #18 Nov 12 2021 00:55:34
%S A117939 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,
%T A117939 -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,
%U A117939 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
%N A117939 Triangle related to powers of 3 partitions of n.
%H A117939 G. C. Greubel, <a href="/A117939/b117939.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A117939 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.
%F A117939 T(n, k) mod 2 = A117944(n,k).
%F A117939 T(n, 0) = A059151(n).
%F A117939 T(n, 1) = A117946(n).
%F A117939 Sum_{k=0..n} T(n, k) = A117940(n).
%F A117939 Matrix square of triangle A117947. Matrix log is the integer triangle A120854. - _Paul D. Hanna_, Jul 08 2006
%e A117939 Triangle begins
%e A117939   1;
%e A117939   2,  1;
%e A117939   1, -2, 1;
%e A117939   2,  0, 0,  1;
%e A117939   4,  2, 0,  2,  1;
%e A117939   2, -4, 2,  1, -2,  1;
%e A117939   1,  0, 0, -2,  0,  0, 1;
%e A117939   2,  1, 0, -4, -2,  0, 2,  1;
%e A117939   1, -2, 1, -2,  4, -2, 1, -2, 1;
%t A117939 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 *)
%o A117939 (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
%o A117939 (Sage)
%o A117939 def A117939(n, k): return sum(jacobi_symbol(binomial(n, j), 3)*jacobi_symbol(binomial(n-j, k), 3) for j in (0..n))
%o A117939 flatten([[A117939(n, k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Oct 29 2021
%Y A117939 Cf. A059151, A117940, A117944, A117946.
%Y A117939 Cf. A120854 (matrix log), A117941 (inverse), A117947 (matrix square-root).
%K A117939 easy,sign,tabl
%O A117939 0,2
%A A117939 _Paul Barry_, Apr 05 2006