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A111940
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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.
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3
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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, 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, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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FORMULA
| 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).
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EXAMPLE
| Triangle P begins:
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;
0,0,0,0,0,0,-1,-1,1; ...
where P^-1 shifts columns left and up one place:
1;
-1,1;
0,1,1;
0,-1,-1,1;
0,0,0,1,1;
0,0,0,-1,-1,1; ...
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PROG
| (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]))}
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CROSSREFS
| Cf. A111941 (matrix log), A111942, A110503 (variant).
Sequence in context: A143142 A174856 A175608 * A129572 A070950 A071031
Adjacent sequences: A111937 A111938 A111939 * A111941 A111942 A111943
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KEYWORD
| sign,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2005
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