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A110503
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Triangle, read by rows, which shifts one column left under matrix inverse.
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8
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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, -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, 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
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OFFSET
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0,8
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COMMENTS
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The unsigned columns of the matrix logarithm of this triangle are all equal to A110504.
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LINKS
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FORMULA
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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.
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).
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EXAMPLE
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Triangle begins:
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, -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; ...
The matrix inverse drops the first column:
1;
-1, 1;
-2, 1, 1;
-1, 1, -1, 1;
-1, 1, -2, 1, 1;
-1, 1, -1, 1, -1, 1; ...
The matrix logarithm equals:
0;
1/1!, 0;
3/2!, -1/1!, 0;
7/3!, -3/2!, 1/1!, 0;
30/4!, -7/3!, 3/2!, -1/1!, 0;
144/5!, -30/4!, 7/3!, -3/2!, 1/1!, 0;
876/6!, -144/5!, 30/4!, -7/3!, 3/2!, -1/1!, 0; ...
unsigned columns of which all equal A110505.
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PROG
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(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]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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