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Search: id:A111940
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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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+0
4
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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; listen)
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OFFSET
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0,1
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FORMULA
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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
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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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PROGRAM
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(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
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Cf. A111941 (matrix log), A111942, A110503 (variant).
Sequence in context: A127244 A144778 A143142 this_sequence A129572 A070950 A141679
Adjacent sequences: A111937 A111938 A111939 this_sequence A111941 A111942 A111943
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KEYWORD
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sign,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2005
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