cp's OEIS Frontend

A110503 Triangle, read by rows, which shifts one column left under matrix inverse.

%I A110503 #10 Jan 17 2018 03:23:41
%S A110503 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 A110503 -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 A110503 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 A110503 Triangle, read by rows, which shifts one column left under matrix inverse.
%C A110503 The unsigned columns of the matrix logarithm of this triangle are all equal to A110504.
%F A110503 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 A110503 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 A110503 Triangle begins:
%e A110503   1;
%e A110503   1,  1;
%e A110503   1, -1,  1;
%e A110503   1, -2,  1,  1;
%e A110503   1, -1,  1, -1,  1;
%e A110503   1, -1,  1, -2,  1,  1;
%e A110503   1, -1,  1, -1,  1, -1,  1;
%e A110503   1, -1,  1, -1,  1, -2,  1,  1;
%e A110503   1, -1,  1, -1,  1, -1,  1, -1,  1;
%e A110503   1, -1,  1, -1,  1, -1,  1, -2,  1,  1; ...
%e A110503 The matrix inverse drops the first column:
%e A110503    1;
%e A110503   -1,  1;
%e A110503   -2,  1,  1;
%e A110503   -1,  1, -1,  1;
%e A110503   -1,  1, -2,  1,  1;
%e A110503   -1,  1, -1,  1, -1,  1; ...
%e A110503 The matrix logarithm equals:
%e A110503      0;
%e A110503     1/1!,     0;
%e A110503     3/2!,   -1/1!,   0;
%e A110503     7/3!,   -3/2!,  1/1!,   0;
%e A110503    30/4!,   -7/3!,  3/2!, -1/1!,  0;
%e A110503   144/5!,  -30/4!,  7/3!, -3/2!, 1/1!,   0;
%e A110503   876/6!, -144/5!, 30/4!, -7/3!, 3/2!, -1/1!, 0; ...
%e A110503 unsigned columns of which all equal A110505.
%o A110503 (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 A110503 Cf. A110504 (matrix log), A110505 (column 0 of log).
%Y A110503 Cf. A111940 (variant).
%K A110503 sign,tabl
%O A110503 0,8
%A A110503 _Paul D. Hanna_, Jul 23 2005