A117906 - OEIS

%I #6 Oct 21 2021 01:30:42

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

%T 0,0,-1,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,

%U -1,1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,1

%N Inverse of number triangle A117904.

%C Row sums are (1, 0, 1, 0, 0, 0, ...) with g.f. 1 + x^2.

%C Diagonal sums are A117907.

%H G. C. Greubel, <a href="/A117906/b117906.txt">Rows n = 0..50 of the triangle, flattened</a>

%F G.f.: (1 -x*(1-y) +x^2*y^2 -x^3*y -x^5*y^2)/(1-x^3*y^3).

%e Triangle begins

%e    1;

%e   -1,  1;

%e    0,  0,  1;

%e    0, -1,  0,  1;

%e    0,  0,  0, -1,  1;

%e    0,  0, -1,  0,  0,  1;

%e    0,  0,  0,  0, -1,  0,  1;

%e    0,  0,  0,  0,  0,  0, -1,  1;

%e    0,  0,  0,  0,  0, -1,  0,  0,  1;

%e    0,  0,  0,  0,  0,  0,  0, -1,  0,  1;

%e    0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1;

%e    0,  0,  0,  0,  0,  0,  0,  0, -1,  0,  0,  1;

%e    0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  0,  1;

%e    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1, 1;

%e    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  0, 0, 1;

%t M[n_, k_]:= M[n, k]= If[k>n, 0, If[Abs[JacobiSymbol[Binomial[n, 2], 3] - JacobiSymbol[Binomial[k, 2], 3]]==0, 1, 0]];

%t m:= m= With[{q=20}, 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 20 2021 *)

%Y Cf. A117904, A117907.

%K easy,sign,tabl

%O 0,1

%A _Paul Barry_, Apr 01 2006