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A126595
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Triangle read by rows: T(0,0)=1; for n>=1, 0<=k<=n, T(n,k) is the coefficient of x^k in the characteristic polynomial (-x)^n+... of the n X n matrix M(n)S(n), where M(n) is the n X n matrix with 0's on the diagonal and 1's elsewhere and S(n) is the n X n matrix whose (i,j) term is 0 for j=i, (-1)^(i+j) for i>j and (-1)^(i+j+1) for i<j.
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0
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1, 0, -1, -1, 0, 1, 0, -3, 0, -1, -3, 0, 2, 0, 1, 0, -5, 0, -10, 0, -1, -5, 0, -5, 0, 9, 0, 1, 0, -7, 0, -35, 0, -21, 0, -1, -7, 0, -28, 0, 14, 0, 20, 0, 1, 0, -9, 0, -84, 0, -126, 0, -36, 0, -1, -9, 0, -75, 0, -42, 0, 90, 0, 35, 0, 1, 0, -11, 0, -165, 0, -462, 0, -330, 0, -55, 0, -1, -11, 0, -154, 0, -297, 0, 132, 0, 275, 0, 54, 0, 1, 0, -13, 0
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OFFSET
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0,8
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COMMENTS
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Sum of terms in row 2n (n>=1) is 0. Sum of the absolute values of the terms in row 2n is C(2n,n) (A000984). All terms in row 2n-1 are nonpositive. Their sum is -4^(n-1). M(2n-1)S(2n-1)=-S(2n-1)
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LINKS
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EXAMPLE
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M(4)=[0,1,1,1/1,0,1,1/1,1,0,1/1,1,1,0], S(4)=[0,1,-1,1/-1,0,1,-1/1,-1,0,1/-1,1,-1,0], M(4)S(4)=[ -1,0,0,0/0,1,-2,2/-2,2,-1,0/0,0,0,1]; char. poly. of M(4)S(4) is x^4 + 2x^2 - 3, yielding row 4 of the triangle: -3,0,2,0,1.
Triangle starts:
1;
0,-1;
-1,0,1;
0,-3,0,-1;
-3,0,2,0,1;
0,-5,0,-10,0,-1
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MAPLE
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with(linalg): m:=proc(i, j) if i=j then 0 else 1 fi end: s:=proc(i, j) if i=j then 0 elif i>j then (-1)^(i+j) else (-1)^(i+j+1) fi end: for n from 1 to 14 do f[n]:=(-1)^n*sort(expand(charpoly(multiply(matrix(n, n, m), matrix(n, n, s)), x))) od: 1; for n from 1 to 14 do seq(coeff(f[n], x, j), j=0..n) od; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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