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A142706
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Coefficients of derivatives of Eulerian number polynomials (A008292): p(x,n)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; p'(x,n)=(d/dx)p{x,n).
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0
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1, 4, 2, 11, 22, 3, 26, 132, 78, 4, 57, 604, 906, 228, 5, 120, 2382, 7248, 4764, 600, 6, 247, 8586, 46857, 62476, 21465, 1482, 7, 502, 29216, 264702, 624760, 441170, 87648, 3514, 8, 1013, 95680, 1365576, 5241416, 6551770, 2731152, 334880, 8104, 9
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Row sums are: A001286 ( Lah numbers: (n-1)*n!/2. );
{0, 1, 6, 36, 240, 1800, 15120, 141120, 1451520, 16329600}.
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FORMULA
| p(x,n)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; p'(x,n)=(d/dx)p{x,n); t(n,m)=Coefficients(p'(x,n)).
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EXAMPLE
| {1},
{4, 2},
{11, 22, 3},
{26, 132, 78, 4},
{57, 604, 906, 228, 5},
{120, 2382, 7248, 4764, 600, 6},
{247, 8586, 46857, 62476, 21465, 1482, 7},
{502, 29216, 264702, 624760, 441170, 87648, 3514, 8},
{1013, 95680, 1365576, 5241416, 6551770, 2731152, 334880, 8104, 9}
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MATHEMATICA
| t[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; Table[D[Sum[t[n, k]*x^k, {k, 0, n - 1}], x], {n, 1, 10}]; Table[CoefficientList[D[Sum[t[n, k]*x^k, {k, 0, n - 1}], x], x], {n, 1, 10}]; Flatten[%]
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CROSSREFS
| Cf. A008292, A001286.
Sequence in context: A185878 A182870 A094406 * A092952 A010318 A188134
Adjacent sequences: A142703 A142704 A142705 * A142707 A142708 A142709
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Sep 24 2008
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