This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A111940 #9 Jan 17 2018 03:23:51
%S A111940 1,1,1,-1,-1,1,0,0,1,1,0,0,-1,-1,1,0,0,0,0,1,1,0,0,0,0,-1,-1,1,0,0,0,
%T A111940 0,0,0,1,1,0,0,0,0,0,0,-1,-1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,-1,
%U A111940 -1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,-1,-1,1
%N A111940 Triangle P, read by rows, that satisfies [P^-1](n,k) = P(n+1,k+1) for n >= k >= 0, with P(k,k)=1 and P(k+1,1)=P(k+1,0) for k >= 0, where [P^-1] denotes the matrix inverse of P.
%F A111940 The g.f. of column k of matrix power P^m (ignoring leading zeros) is:
%F A111940 cos(m*arccos(1-x^2/2)) + (-1)^k * sin(m*arccos(1-x^2/2)) * (1-x/2) / sqrt(1-x^2/4).
%e A111940 Triangle P begins:
%e A111940 1;
%e A111940 1, 1;
%e A111940 -1, -1, 1;
%e A111940 0, 0, 1, 1;
%e A111940 0, 0, -1, -1, 1;
%e A111940 0, 0, 0, 0, 1, 1;
%e A111940 0, 0, 0, 0, -1, -1, 1;
%e A111940 0, 0, 0, 0, 0, 0, 1, 1;
%e A111940 0, 0, 0, 0, 0, 0, -1, -1, 1; ...
%e A111940 where P^-1 shifts columns left and up one place:
%e A111940 1;
%e A111940 -1, 1;
%e A111940 0, 1, 1;
%e A111940 0, -1, -1, 1;
%e A111940 0, 0, 0, 1, 1;
%e A111940 0, 0, 0, -1, -1, 1; ...
%o A111940 (PARI) {P(n,k,q=-1) = local(A=Mat(1),B); if(n<k||k<0,0, for(m=1,n+1, B = matrix(m,m); for(i=1,m, for(j=1,i, if(j==i, B[i,j]=1, if(j==1, B[i,j]=(A^q)[i-1,1], B[i,j] = (A^q)[i-1,j-1]));)); A=B); return(A[n+1,k+1]))}
%o A111940 for(n=0,16,for(k=0,n, print1(P(n,k,-1),", "));print(""))
%Y A111940 Cf. A111941 (matrix log), A111942, A110503 (variant).
%K A111940 sign,tabl
%O A111940 0,1
%A A111940 _Paul D. Hanna_, Aug 23 2005