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A140682
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A triangle of coefficients of the difference of the GCD and Binomial in a polynomials: p(x,n)=Sum[GCd(n,i)-Binmomial(n,i))*x^i,{i,0,n}].
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0
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0, 1, -1, 1, 2, -2, -2, 2, 3, -3, -4, -3, 3, 4, -4, -9, -9, -4, 4, 5, -5, -13, -17, -13, -5, 5, 6, -6, -20, -34, -34, -20, -6, 6, 7, -7, -26, -55, -66, -55, -26, -7, 7, 8, -8, -35, -81, -125, -125, -81, -35, -8, 8, 9, -9, -43, -119, -208, -247, -208, -119, -43, -9, 9
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Row sums are:
{0, 1, 0, -4, -18, -43, -108, -228, -482, -987};
This is a method of producing symmetrical polynomials that is combinatorial related.
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FORMULA
| p(x,n)=Sum[GCd(n,i)-Binmomial(n,i))*x^i,{i,0,n}]; a(n,m)=Coefficiencts(p(x,n))
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EXAMPLE
| {0},
{1, -1, 1},
{2, -2, -2, 2},
{3, -3, -4, -3, 3},
{4, -4, -9, -9, -4, 4},
{5, -5, -13, -17, -13, -5, 5},
{6, -6, -20, -34, -34, -20, -6, 6},
{7, -7, -26, -55, -66, -55, -26, -7, 7},
{8, -8, -35, -81, -125, -125, -81, -35, -8, 8},
{9, -9, -43, -119, -208, -247, -208, -119, -43, -9, 9}
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MATHEMATICA
| Clear[p, x, n] p[x_, n_] = Sum[(GCD[n, i] - Binomial[n, i])*x^i, {i, 0, n}]; Table[ExpandAll[p[x, n]], {n, 1, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 1, 10}]; Flatten[a]
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CROSSREFS
| Cf. A109004.
Sequence in context: A029084 A032229 A024366 * A049317 A134544 A106161
Adjacent sequences: A140679 A140680 A140681 * A140683 A140684 A140685
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KEYWORD
| tabl,uned,sign
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jul 11 2008
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