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%I #10 Jan 17 2018 03:23:41
%S 1,1,1,1,-1,1,1,-2,1,1,1,-1,1,-1,1,1,-1,1,-2,1,1,1,-1,1,-1,1,-1,1,1,
%T -1,1,-1,1,-2,1,1,1,-1,1,-1,1,-1,1,-1,1,1,-1,1,-1,1,-1,1,-2,1,1,1,-1,
%U 1,-1,1,-1,1,-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,1,-2,1,1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,1,-1,1,-1,1,-1,1
%N Triangle, read by rows, which shifts one column left under matrix inverse.
%C The unsigned columns of the matrix logarithm of this triangle are all equal to A110504.
%F T(n, k) = +1 when k == 0 (mod 2), T(n, k)=-1 when k == 1 (mod 2), except for T(k+2, k) = -2 when k == 1 (mod 2) and T(n, n) = 1.
%F G.f. for column k of matrix power A110503^m (ignoring leading zeros): cos(m*arccos(1-x^2/2)) + (-1)^k*sin(m*arccos(1-x^2/2))*(1-x/2)/sqrt(1-x^2/4)*(1+x)/(1-x).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, -1, 1;
%e 1, -2, 1, 1;
%e 1, -1, 1, -1, 1;
%e 1, -1, 1, -2, 1, 1;
%e 1, -1, 1, -1, 1, -1, 1;
%e 1, -1, 1, -1, 1, -2, 1, 1;
%e 1, -1, 1, -1, 1, -1, 1, -1, 1;
%e 1, -1, 1, -1, 1, -1, 1, -2, 1, 1; ...
%e The matrix inverse drops the first column:
%e 1;
%e -1, 1;
%e -2, 1, 1;
%e -1, 1, -1, 1;
%e -1, 1, -2, 1, 1;
%e -1, 1, -1, 1, -1, 1; ...
%e The matrix logarithm equals:
%e 0;
%e 1/1!, 0;
%e 3/2!, -1/1!, 0;
%e 7/3!, -3/2!, 1/1!, 0;
%e 30/4!, -7/3!, 3/2!, -1/1!, 0;
%e 144/5!, -30/4!, 7/3!, -3/2!, 1/1!, 0;
%e 876/6!, -144/5!, 30/4!, -7/3!, 3/2!, -1/1!, 0; ...
%e unsigned columns of which all equal A110505.
%o (PARI) T(n,k)=matrix(n+1,n+1,r,c,if(r>=c, if(r==c || c%2==1,1,if(r%2==0 && r==c+2,-2,-1))))[n+1,k+1]
%Y Cf. A110504 (matrix log), A110505 (column 0 of log).
%Y Cf. A111940 (variant).
%K sign,tabl
%O 0,8
%A _Paul D. Hanna_, Jul 23 2005
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