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A141720
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A coefficient triangle sequence of a new infinite sum polynomial set: p(x,n)=((-1 + 2*x)^(n + 1)/(-1 + x)^(n + 1))*Sum[k^n*(1 - x)^(n - k)*x^(k), {k, 0, Infinity}]; t(n,m)=Coefficients(p(x,n)).
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1
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1, 0, 1, 0, 1, 0, 1, 2, -2, 0, 1, 8, -8, 0, 1, 22, -6, -32, 16, 0, 1, 52, 84, -272, 136, 0, 1, 114, 606, -1168, -96, 816, -272, 0, 1, 240, 2832, -2176, -8832, 11904, -3968, 0, 1, 494, 11122, 11072, -83360, 71168, 13312, -31744, 7936, 0, 1, 1004, 39772, 148592, -472760, -17152, 831232, -707584, 176896
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| Row sums are one.
This set of polynomials comes from looking for Bernstein like polynomial
as infinite sums of the Eulerian number type.
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FORMULA
| p(x,n)=((-1 + 2*x)^(n + 1)/(-1 + x)^(n + 1))*Sum[k^n*(1 - x)^(n - k)*x^(k), {k, 0, Infinity}]; t(n,m)=Coefficients(p(x,n)).
p(x,n)=Sum(eulerian(n,k)*Sum((-1)^l*(n-l+1)*(2-x)^l*binomial(l+1,k),l=0..n),k=0..n) [From Mourad Rahmani (mrahmani(AT)usthb.dz), Jul 22 2010]
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EXAMPLE
| {1},
{0, 1},
{0, 1},
{0, 1, 2, -2},
{0, 1, 8, -8},
{0, 1, 22, -6, -32, 16},
{0, 1, 52, 84, -272, 136},
{0, 1, 114, 606, -1168, -96, 816, -272},
{0, 1, 240, 2832, -2176, -8832, 11904, -3968},
{0, 1, 494, 11122, 11072, -83360, 71168, 13312, -31744, 7936},
{0, 1, 1004, 39772, 148592, -472760, -17152, 831232, -707584, 176896}
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MATHEMATICA
| Table[CoefficientList[FullSimplify[ExpandAll[((-1 + 2*x)^(n + 1)/(-1 +x)^(n + 1))*Sum[k^n*(1 - x)^(n - k)*x^(k), {k, 0, Infinity}]]], x], {n, 0, 10}]; Flatten[%]
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CROSSREFS
| Sequence in context: A065600 A029583 A011289 * A103272 A065710 A089799
Adjacent sequences: A141717 A141718 A141719 * A141721 A141722 A141723
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KEYWORD
| uned,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 11 2008
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