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Search: id:A141693
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| A141693 |
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A triangle of coefficients made from the derivative of Eulerian-Bernstein's polynomials at the midpoint t=1/2: p(x,n,m)=Eulerian[n,m]*x^m*(1-x)^(n-m); t(n,m)=d/dx*p(x,n,m) such that x=1/2. |
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+0
1
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| 0, -1, 1, -2, 0, 2, -3, -4, 1, 3, -4, -22, 0, 2, 4, -5, -78, -66, 26, 3, 5, -6, -228, -604, 0, 114, 4, 6, -7, -600, -3573, -2416, 1191, 360, 5, 7, -8, -1482, -17172, -31238, 0, 8586, 988, 6, 8, -9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9, -10, -8104, -287040, -1820768, -2620708, 0, 910384
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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The row sums are:
{0, 0, 0, -3, -20, -115, -714, -5033, -40312, -362871, -3628790}.
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REFERENCES
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Weisstein, Eric W. "Bernstein Polynomial." http : // mathworld.wolfram.com/BernsteinPolynomial.html
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FORMULA
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p(x,n,m)=Eulerian[n,m]*x^m*(1-x)^(n-m); t(n,m)=d/dx*p(x,n,m) such that x=1/2.
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EXAMPLE
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{0},
{-1, 1},
{-2, 0, 2},
{-3, -4, 1, 3},
{-4, -22, 0, 2, 4},
{-5, -78, -66, 26, 3, 5},
{-6, -228, -604, 0, 114, 4, 6},
{-7, -600, -3573, -2416,1191, 360, 5, 7},
{-8, -1482, -17172, -31238, 0, 8586, 988, 6, 8},
{-9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9},
{-10, -8104, -287040, -1820768, -2620708, 0, 910384, 191360, 6078, 8, 10}
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MATHEMATICA
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Clear[BernstenB, n, i, t]; BernsteinB[0, n_, 0] := 1; BernsteinB[n_, n_, t_] := t^n; BernsteinB[i_, n_, t_] := Sum[(-1)^j Binomial[n + 1, j](i + 1 - j)^n, {j, 0, i + 1}]t^i(1 - t)^(n - i); Table[Table[2^(n - 1)*(D[BernsteinB[i, n, t], t] /. t -> 1/2), {i, 0, n}], {n, 0, 10}]; Flatten[%]
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CROSSREFS
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Cf. A123948.
Sequence in context: A003987 A141692 A063180 this_sequence A028376 A059235 A072514
Adjacent sequences: A141690 A141691 A141692 this_sequence A141694 A141695 A141696
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KEYWORD
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uned,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 09 2008
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