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%I #13 Feb 21 2024 08:18:58
%S 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,
%T 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,
%U 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
%N Triangle related to powers of 3 partitions of n.
%C Row sums are A039966.
%C Inverse of A117944.
%H G. C. Greubel, <a href="/A117945/b117945.txt">Rows n = 0..50 of the triangle, flattened</a>
%e Triangle begins
%e 1;
%e 0, 1;
%e -1, 0, 1;
%e 0, 0, 0, 1;
%e 0, 0, 0, 0, 1;
%e 0, 0, 0, -1, 0, 1;
%e -1, 0, 0, 0, 0, 0, 1;
%e 0, -1, 0, 0, 0, 0, 0, 1;
%e 1, 0, -1, 0, 0, 0, -1, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1;
%t M[n_, k_]:= M[n, k] = If[k>n, 0, Mod[Sum[JacobiSymbol[Binomial[n, j], 3]*JacobiSymbol[Binomial[n-j, k], 3], {j,0,n}], 2], 0];
%t m:= m= With[{q=60}, Table[M[n, k], {n,0,q}, {k,0,q}]];
%t T[n_, k_]:= Inverse[m][[n+1, k+1]];
%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 29 2021 *)
%Y Cf. A039966, A117944.
%K sign,tabl
%O 0,1
%A _Paul Barry_, Apr 05 2006