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A118400
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Triangle T, read by rows, where all columns of T are different and yet all columns of the matrix square T^2 (A118401) are equal; a signed version of triangle A087698.
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3
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1, 1, -1, 1, 0, 1, -1, -1, -1, -1, 1, 2, 2, 2, 1, -1, -3, -4, -4, -3, -1, 1, 4, 7, 8, 7, 4, 1, -1, -5, -11, -15, -15, -11, -5, -1, 1, 6, 16, 26, 30, 26, 16, 6, 1, -1, -7, -22, -42, -56, -56, -42, -22, -7, -1, 1, 8, 29, 64, 98, 112, 98, 64, 29, 8, 1, -1, -9, -37, -93, -162, -210, -210, -162, -93, -37, -9, -1
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OFFSET
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0,12
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COMMENTS
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Matrix inverse equals A118404. Row sums equal A084633. Signed version of: A087698 = maximum number of Boolean inputs at Hamming distance 2 for symmetric Boolean functions. This is an example of the fact that special matrices (cf. A118401) can have more than 2 signed matrix square-roots if the main diagonal is allowed to be signed.
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LINKS
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FORMULA
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G.f.: A(x,y) = (1+2*x+2*x^2)/(1+x+x*y). G.f. of column k = (-1)^k*(1+2*x+2*x^2)/(1+x)^(k+1) for k>=0. T(n,k) = (-1)^n*[C(n,k) - 2*C(n-2,k-1)] for n>=k>=0 except that T(1,0)=1.
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EXAMPLE
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Triangle T begins:
1;
1,-1;
1, 0, 1;
-1,-1,-1,-1;
1, 2, 2, 2, 1;
-1,-3,-4,-4,-3,-1;
1, 4, 7, 8, 7, 4, 1;
-1,-5,-11,-15,-15,-11,-5,-1;
1, 6, 16, 26, 30, 26, 16, 6, 1;
-1,-7,-22,-42,-56,-56,-42,-22,-7,-1;
1, 8, 29, 64, 98, 112, 98, 64, 29, 8, 1;
-1,-9,-37,-93,-162,-210,-210,-162,-93,-37,-9,-1; ...
1;
0, 1;
2, 0, 1;
-2, 2, 0, 1;
4,-2, 2, 0, 1;
-6, 4,-2, 2, 0, 1;
8,-6, 4,-2, 2, 0, 1;
-10, 8,-6, 4,-2, 2, 0, 1;
12,-10, 8,-6, 4,-2, 2, 0, 1; ...
in which all columns are equal.
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PROG
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(PARI) T(n, k)=polcoeff(polcoeff((1+2*x+2*x^2)/(1+x+x*y+x*O(x^n)), n, x)+y*O(y^k), k, y)
(PARI) T(n, k)=if(n==1&k==0, 1, (-1)^n*(binomial(n, k)-2*binomial(n-2, 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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