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Search: id:A142175
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| A142175 |
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Quadratic Bezier of Pascal, Eulerian numbers and A060187 triangular sequences: p(x,n,y)=(y^2 (1 + x)^n + 2^n (1 - y)^2 (1 - x)^(n + 1) LerchPhi[x, -n, 1/2] + 2* (1 - y) y (1 - x)^(n + 2)*PolyLog[ -(n + 1), x]/x);y=-1/2. |
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| 1, 1, 1, 1, 8, 1, 1, 36, 36, 1, 1, 133, 420, 133, 1, 1, 449, 3334, 3334, 449, 1, 1, 1446, 21939, 49364, 21939, 1446, 1, 1, 4534, 130044, 560957, 560957, 130044, 4534, 1, 1, 13991, 724222, 5459561, 10284514, 5459561, 724222, 13991, 1, 1, 42747, 3880014
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row sums are:
{1, 2, 10, 74, 688, 7568, 96136, 1391072, 22680064, 412594688, 8300880256}.
This Bezier solution is probably an upper bound estimate.
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FORMULA
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p(x,n,y)=(y^2 (1 + x)^n + 2^n (1 - y)^2 (1 - x)^(n + 1) LerchPhi[x, -n, 1/2] + 2* (1 - y) y (1 - x)^(n + 2)*PolyLog[ -(n + 1), x]/x);y=-1/2; t(n,m)=coefficients(p(x,n,-1/2)).
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EXAMPLE
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{1},
{1, 1},
{1, 8, 1},
{1, 36, 36, 1},
{1, 133, 420, 133, 1},
{1, 449, 3334, 3334, 449, 1},
{1, 1446, 21939, 49364, 21939, 1446, 1},
{1, 4534, 130044, 560957, 560957, 130044, 4534, 1},
{1, 13991, 724222, 5459561, 10284514, 5459561, 724222, 13991, 1},
{1, 42747, 3880014, 48160170, 154214412, 154214412, 48160170, 3880014, 42747, 1},
{1, 129784, 20282373, 397871832, 2025970266, 3412371744, 2025970266, 397871832, 20282373, 129784, 1}
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MATHEMATICA
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p[x_, n_, y_] = (y^2 (1 + x)^n + 2^n (1 - y)^2 (1 - x)^(n + 1) LerchPhi[x, -n, 1/2] + 2* (1 - y) y (1 - x)^(n + 2)*PolyLog[ -(n + 1), x]/x); a = Table[CoefficientList[FullSimplify[Expand[p[x, n, y]]], x], {n, 0, 10}] /. y -> -1/2; Flatten[a]
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CROSSREFS
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Cf. A060187.
Sequence in context: A157148 A154335 A142467 this_sequence A142597 A156137 A152972
Adjacent sequences: A142172 A142173 A142174 this_sequence A142176 A142177 A142178
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Sep 16 2008
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