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A143505
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The PolyLog version of Eulerian numbers (A008292) with the McMullen's transform substitution:x->x+1/x: p(x,n) = (1 - x)^(n + 1)*PolyLog[ -n, x]/x; q(x,n)=x^(n-1)*p(x+1/x,n).
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0
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1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 11, 14, 23, 14, 11, 1, 1, 26, 70, 104, 139, 104, 70, 26, 1, 1, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1, 1, 120, 1197, 3016, 5970, 8568, 9549, 8568, 5970, 3016, 1197, 120, 1, 1, 247, 4300, 17101, 37105, 70474, 90069, 107241, 90069
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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COMMENTS
| Row sums are:
{1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563}.
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FORMULA
| p(x,n) = (1 - x)^(n + 1)*PolyLog[ -n, x]/x; q(x,n)=x^(n-1)*p(x+1/x,n); t(n,m)=Coefficients(q(x,n)).
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EXAMPLE
| {1},
{1, 1, 1},
{1, 4, 3, 4, 1},
{1, 11, 14, 23, 14, 11, 1},
{1, 26, 70, 104, 139, 104, 70, 26, 1},
{1, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1},
{1, 120, 1197, 3016, 5970, 8568, 9549, 8568, 5970, 3016, 1197, 120, 1},
{1, 247, 4300, 17101, 37105, 70474, 90069, 107241, 90069, 70474, 37105, 17101, 4300, 247, 1},
{1, 502, 14616, 91748, 243866, 539946, 858544, 1165114, 1258587, 1165114, 858544, 539946, 243866, 91748, 14616, 502, 1},
{1, 1013, 47849, 463296, 1645270, 4069870, 8011686, 12173864, 16144655, 17132555, 16144655, 12173864, 8011686, 4069870, 1645270, 463296, 47849, 1013, 1}
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MATHEMATICA
| p[x_, n_] = (1 - x)^(n + 1)*PolyLog[ -n, x]/x; Table[FullSimplify[ExpandAll[x^n*p[x + 1/x, n]]], {n, 1, 10}]; Table[CoefficientList[FullSimplify[ExpandAll[x^(n - 1)*p[x + 1/x, n]]], x], {n, 1, 10}]; Flatten[%]
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CROSSREFS
| Cf. A008292.
Sequence in context: A070511 A066340 A195597 * A170987 A196826 A204819
Adjacent sequences: A143502 A143503 A143504 * A143506 A143507 A143508
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 25 2008
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