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Search: id:A142147
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| A142147 |
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A PolyLog functional polynomial coefficient triangular sequence: If[n == 0, 1, (1 + 2 (-1 + x) x)^(n + 1)*PolyLog[ -n, 2*x*(1 - x)]/(2*x)]. |
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+0
1
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| 1, 1, -1, 1, 1, -4, 2, 1, 7, -12, -4, 12, -4, 1, 21, 0, -102, 100, 4, -32, 8, 1, 51, 160, -532, -24, 904, -672, 48, 80, -16, 1, 113, 980, -1094, -5128, 8760, -736, -6224, 3920, -432, -192, 32, 1, 239, 4284, 5276, -43964, 19764, 90272, -114080, 19824, 36304, -20800, 2496, 448, -64, 1, 493, 16184, 73930, -198388
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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Row sums are zero except for one:
{1, 7, 61, 679, 9445, 158095, 3088765, 68958295, 1731875605, 48328686175}.
The polynomial 2*x*(1-x) is the middle quadratic Bezier/ Berstein function:
{x^2,2*x*(1-x),1-2*x+x^2}. The PolyLog logical conjugate is (1 + 2 (-1 + x) x).
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FORMULA
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p(x,n)=If[n == 0, 1, (1 + 2 (-1 + x) x)^(n + 1)*PolyLog[ -n, 2*x*(1 - x)]/(2*x)]; t(n,m)=corefficients(p(x,n)).
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EXAMPLE
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{1},
{1, -1},
{1, 1, -4, 2},
{1, 7, -12, -4, 12, -4},
{1, 21, 0, -102,100, 4, -32, 8},
{1, 51, 160, -532, -24, 904, -672, 48, 80, -16},
{1, 113, 980, -1094, -5128, 8760, -736, -6224, 3920, -432, -192, 32},
{1, 239, 4284, 5276, -43964, 19764, 90272, -114080, 19824, 36304, -20800, 2496, 448, -64},
{1, 493, 16184, 73930, -198388, -379604, 1190784, -423976, -1167056, 1261040, -278144, -185632, 103488, -12224, -1024, 128}, {1, 1003, 56424, 531580, -149152, -5494912, 6160896, 11587664, -24855200, 8928224, 12191872, -12333248, 2978304, 832000, -492544, 55040, 2304, -256},
{1, 2025,187308, 3069482, 6973600, -41239136, -27365184, 225152320, -170749024, -242500704, 434455680, -162960832, -106377216, 109758464, -27333632, -3104768, 2272512, -236288, -5120, 512}
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MATHEMATICA
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Clear[p, x, n, m]; p[x_, n_] = If[n == 0, 1, (1 + 2 (-1 + x) x)^(n + 1)*PolyLog[ -n, 2*x*(1 - x)]/(2*x)]; Table[FullSimplify[Expand[p[x, n]]], {n, 1, 10}]; Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 1, 10}]; Flatten[%]
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CROSSREFS
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Cf. A008292, A123125.
Sequence in context: A039962 A046741 A136249 this_sequence A142073 A135294 A117016
Adjacent sequences: A142144 A142145 A142146 this_sequence A142148 A142149 A142150
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KEYWORD
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sign,uned
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Sep 15 2008
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