%I A118404
%S A118404 1,1,1,1,0,1,1,1,1,1,1,0,0,2,1,1,1,0,2,3,1,1,0,1,2,5,4,1,1,1,1,3,7,9,5,
%T A118404 1,1,0,0,4,10,16,14,6,1,1,1,0,4,14,26,30,20,7,1,1,0,1,4,18,40,56,50,27,
%U A118404 8,1,1,1,1,5,22,58,96,106,77,35,9,1
%V A118404 1,1,-1,-1,0,1,-1,1,-1,-1,1,0,0,2,1,1,-1,0,-2,-3,-1,-1,0,1,2,5,4,1,-1,
               1,-1,-3,-7,-9,-5,
%W A118404 -1,1,0,0,4,10,16,14,6,1,1,-1,0,-4,-14,-26,-30,-20,-7,-1,-1,0,1,4,18,40,
               56,50,27,8,1,
%X A118404 -1,1,-1,-5,-22,-58,-96,-106,-77,-35,-9,-1
%N A118404 Triangle T, read by rows, where all columns of T are different and yet 
               all columns of the matrix square T^2 (A118407) are equal; also equals 
               the matrix inverse of triangle A118400.
%F A118404 G.f.: A(x,y) = (1+x)^2/(1+x^2)/(1+x+x*y). G.f. of column k = (-1)^k/(1+x^2)/
               (1+x)^(k-1) for k>=0.
%e A118404 Triangle begins:
%e A118404 1;
%e A118404 1,-1;
%e A118404 -1, 0, 1;
%e A118404 -1, 1,-1,-1;
%e A118404 1, 0, 0, 2, 1;
%e A118404 1,-1, 0,-2,-3,-1;
%e A118404 -1, 0, 1, 2, 5, 4, 1;
%e A118404 -1, 1,-1,-3,-7,-9,-5,-1;
%e A118404 1, 0, 0, 4, 10, 16, 14, 6, 1;
%e A118404 1,-1, 0,-4,-14,-26,-30,-20,-7,-1;
%e A118404 -1, 0, 1, 4, 18, 40, 56, 50, 27, 8, 1;
%e A118404 -1, 1,-1,-5,-22,-58,-96,-106,-77,-35,-9,-1;
%e A118404 1, 0, 0, 6, 27, 80, 154, 202, 183, 112, 44, 10, 1; ...
%e A118404 The matrix square is A118407:
%e A118404 1;
%e A118404 0, 1;
%e A118404 -2, 0, 1;
%e A118404 2,-2, 0, 1;
%e A118404 0, 2,-2, 0, 1;
%e A118404 -2, 0, 2,-2, 0, 1;
%e A118404 4,-2, 0, 2,-2, 0, 1;
%e A118404 -6, 4,-2, 0, 2,-2, 0, 1;
%e A118404 4,-6, 4,-2, 0, 2,-2, 0, 1;
%e A118404 6, 4,-6, 4,-2, 0, 2,-2, 0, 1; ...
%e A118404 in which all columns are equal.
%o A118404 (PARI) {T(n,k)=polcoeff(polcoeff((1+x)^2/(1+x^2)/(1+x+x*y +x*O(x^n)),
               n,x)+y*O(y^k),k,y)}
%Y A118404 Cf. A118405 (row sums), A118406 (unsigned row sums), A118407 (matrix 
               square), A118400 (matrix inverse).
%Y A118404 Sequence in context: A109707 A064272 A117479 this_sequence A089339 A127284 
               A120691
%Y A118404 Adjacent sequences: A118401 A118402 A118403 this_sequence A118405 A118406 
               A118407
%K A118404 sign,tabl
%O A118404 0,14
%A A118404 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 27 2006