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A192174
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Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.
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5
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-1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, 0, -1, 0, 1, 0, -3, 0, 0, 0, 1, 1, 0, -4, 0, 2, 0, 2, 0, 1, 0, -5, 0, 5, 0, 2, 0, -1, 1, 0, -6, 0, 9, 0, 0, 0, -3, 0, 1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1, 1, 0, -8, 0, 20, 0, -14, 0, -5, 0, 4, 0
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OFFSET
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0,18
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COMMENTS
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Consider the Catalan triangle A009766 antisymmetrically extended by a mirror along the diagonal (see also A176239):
0, -1, -1, -1, -1, -1, -1, -1,
1, 0, -1, -2, -3, -4, -5, -6,
1, 1, 0, -2, -5, -9, -14, -20,
1, 2, 2, 0, -5, -14, -28, -48,
1, 3, 5, 5, 0, -14, -42, -90,
1, 4, 9, 14, 14, 0, -42, -132,
1, 5, 14, 28, 42, 42, 0, -132,
1, 6, 20, 48, 90, 132, 132, 0.
The rows in this array are essentially the columns of T(n,k).
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LINKS
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FORMULA
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Sum_{k=0..n} T(n,k) = A057079(n-1).
Apparently T(3s,2s-2) = (-1)^(s+1)*A000245(s), s >= 1.
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EXAMPLE
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Triangle begins
-1; # -1
1, 0; # x
1, 0, 1; # x^2+1
1, 0, 0, 0; # x^3
1, 0, -1, 0, -1; # x^4-x^2-1
1, 0, -2, 0, -1, 0;
1, 0, -3, 0, 0, 0, 1;
1, 0, -4, 0, 2, 0, 2, 0;
1, 0, -5, 0, 5, 0, 2, 0, -1;
1, 0, -6, 0, 9, 0, 0, 0, -3, 0;
1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1;
1, 0, -8, 0, 20, 0,-14, 0, -5, 0, 4, 0;
1, 0, -9, 0, 27, 0,-28, 0, 0, 0, 9, 0, -1;
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MAPLE
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p:= proc(n, x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1, x)-procname(n-2, x) fi: end: A192174 := proc(n, k): coeff(p(n, x), x, n-k): end: seq(seq(A192174(n, k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 21 2011
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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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