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Search: id:A147563
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| A147563 |
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A scaled Bernstein-type polynomial set based on A008292 Eulerian numbers: p(x,n)=4*Sum[A0008292[[n]][[m + 1]]*(x/2)^(n - m - 1)*(1 - x/2)^m, {m, 0, n - 1}] |
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+0
1
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| 4, 4, 4, 4, -2, 4, 16, -8, 4, 44, -6, -16, 4, 4, 104, 84, -136, 34, 4, 228, 606, -584, -24, 102, -17, 4, 480, 2832, -1088, -2208, 1488, -248, 4, 988, 11122, 5536, -20840, 8896, 832, -992, 124, 4, 2008, 39772, 74296, -118190, -2144, 51952, -22112, 2764, 4, 4052
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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The row sums are: {4, 4, 6, 12, 30, 90, 315, 1260, 5670, 28350, 155925,...}. The Eulerian number scaling is based on the quadratic Sierpinski number: (1,2*k,1}: to give a Bernstein Sierpinski probability scale of: p=x/2*(k-1). Here k=2 so k-1 is 2; for Pascal it is k=1 or 2^0.
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FORMULA
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p(x,n)=4*Sum[A0008292[[n]][[m + 1]]*(x/2)^(n - m - 1)*(1 - x/2)^m, {m, 0, n - 1}]; t(n,m)=coefficients(t(n,m)).
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EXAMPLE
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{4}, {4}, {4, 4, -2}, {4, 16, -8}, {4, 44, -6, -16, 4}, {4, 104, 84, -136, 34}, {4, 228, 606, -584, -24, 102, -17}, {4, 480, 2832, -1088, -2208, 1488, -248}, {4, 988, 11122, 5536, -20840, 8896, 832, -992, 124}, {4, 2008,39772, 74296, -118190, -2144, 51952, -22112, 2764}, {4, 4052, 134358, 527784, -395820, -590322, 655923, -172608, -19884, 13820, -1382}
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MATHEMATICA
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p[x_, n_] = (1 - x)^(n + 1)*PolyLog[ -n, x]/x; a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 11}]; Table[CoefficientList[FullSimplify[ExpandAll[4*Sum[a[[n]][[m + 1]]*(x/2)^(n - m - 1)*(1 - x/2)^m, {m, 0, n - 1}]]], x], {n, 1, 11}]; Flatten[%]
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CROSSREFS
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Sequence in context: A074803 A046595 A046587 this_sequence A136213 A088848 A088849
Adjacent sequences: A147560 A147561 A147562 this_sequence A147564 A147565 A147566
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KEYWORD
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sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 07 2008
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