%I A111940
%S A111940 1,1,1,1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,1,1,1,0,0,0,0,0,0,1,
%T A111940 1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,
%U A111940 0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,1,1
%V 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,-1,1,0,0,0,0,
               0,0,1,1,0,0,0,0,
%W A111940 0,0,-1,-1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,-1,-1,1,0,0,0,0,0,0,0,
               0,0,0,1,1,0,0,0,
%X A111940 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: 
               cos(m*acos(1-x^2/2))+(-1)^k*sin(m*acos(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]))}
%Y A111940 Cf. A111941 (matrix log), A111942, A110503 (variant).
%Y A111940 Sequence in context: A127244 A144778 A143142 this_sequence A129572 A070950 
               A141679
%Y A111940 Adjacent sequences: A111937 A111938 A111939 this_sequence A111941 A111942 
               A111943
%K A111940 sign,tabl
%O A111940 0,1
%A A111940 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2005