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Triangle related to powers of 3 partitions of n.
5

%I #10 Nov 12 2021 00:53:01

%S 1,0,1,1,0,1,0,0,0,1,0,0,0,0,1,0,0,0,1,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,

%T 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 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 Inverse is A117945.

%C Row sums of inverse are A039966.

%H G. C. Greubel, <a href="/A117944/b117944.txt">Rows n = 0..50 of the triangle, flattened</a>

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

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

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

%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 T[n_, k_]:= Mod[Sum[JacobiSymbol[Binomial[n, j], 3]*JacobiSymbol[Binomial[n-j, k], 3], {j,0,n}], 2];

%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 29 2021 *)

%o (Sage)

%o 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 flatten([[A117944(n, k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Oct 29 2021

%Y Cf. A039966, A117939, A117943, A117945.

%K easy,nonn,tabl

%O 0,1

%A _Paul Barry_, Apr 05 2006