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A140575
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Triangle read by rows: the coefficient of [x^k] of the polynomial 1-(x-1)^n in row n and column k, 0<=k<n.
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1
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0, 2, -1, 0, 2, -1, 2, -3, 3, -1, 0, 4, -6, 4, -1, 2, -5, 10, -10, 5, -1, 0, 6, -15, 20, -15, 6, -1, 2, -7, 21, -35, 35, -21, 7, -1, 0, 8, -28, 56, -70, 56, -28, 8, -1, 2, -9, 36, -84, 126, -126, 84, -36, 9, -1, 0, 10, -45, 120, -210, 252, -210, 120, -45, 10, -1
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OFFSET
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0,2
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COMMENTS
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Row sums are: 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...;
This is the Pascal Triangle A007318 with alternating signs and the leading column of 1's replaced alternatingly by 0 and 2. - R. J. Mathar, Sep 09 2013
With T(0,0) = 1, this is (2, -2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 24 2015
G.f.: (1+2*x-x^2-2*x*y+x^2*y)/((-1+x)*(-x+x*y-1)) -1 - R. J. Mathar, Aug 12 2015
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LINKS
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FORMULA
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T(n,k) = T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = 0, T(1,0) = 2, T(1,1) = -1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, May 24 2015
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EXAMPLE
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0;
2, -1;
0, 2, -1;
2, -3, 3, -1;
0, 4, -6, 4, -1;
2, -5, 10, -10, 5, -1;
0, 6, -15, 20, -15, 6, -1;
2, -7, 21, -35, 35, -21, 7, -1;
0, 8, -28,56, -70, 56, -28, 8, -1;
2, -9, 36, -84, 126, -126, 84, -36, 9, -1;
0, 10, -45, 120, -210, 252, -210, 120, -45, 10, -1;
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MATHEMATICA
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Clear[p] p[x, 0] = 1; p[x, 1] = x - 1; p[x_, n_] := x^n*(1/x^n - (1 - 1/x)^n); a = Table[ExpandAll[p[x, n]], {n, 0, 10}]; b = Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]; Flatten[b]
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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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