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A152249
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Triangle of 4 - restricted Eulerian numbers as polynomials used in exponential data smoothing: m(p,k,x)=((-1)^k*(1 - x)^(p + k)/(k!(p - 1)!))*Sum[(p - 1 + j)!*j^k*x^j/(j!), {j, 0, Infinity}]/x;n=6; t(m,l)=coefficients((-1)^m*m!*M[n, m, x]]]/n
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0
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1, 1, 6, 1, 19, 36, 1, 46, 241, 216, 1, 101, 1091, 2551, 1296, 1, 212, 4182, 18932, 24337, 7776, 1, 435, 14666, 113366, 273141, 217015, 46656, 1, 882, 48783, 600124, 2385999, 3487218, 1845697, 279936, 1, 1777, 156933, 2937109, 17931235, 42397299
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Row sums are: {1, 7, 56, 504, 5040, 55440, 665280, 8648640, 121080960, 1816214400,...}. The sequences A008292, A144696,A144697,A144698,A144699 and this one, form a matrix of polynomials that are used in data smoothing calculations.
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REFERENCES
| Douglas C. Montgomery, Lynwood A, Johnson, Forecasting and Time Series Analysis,McGraw-Hill, New York,1976,page 64.
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FORMULA
| m(p,k,x)=((-1)^k*(1 - x)^(p + k)/(k!(p - 1)!))*Sum[(p - 1 + j)!*j^k*x^j/(j!), {j, 0, Infinity}]/x;n=6;
t(m,l)=coefficients((-1)^m*m!*M[n, m, x]]]/n
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EXAMPLE
| {1},
{1, 6},
{1, 19, 36},
{1, 46, 241, 216},
{1, 101, 1091, 2551, 1296},
{1, 212, 4182, 18932, 24337, 7776},
{1, 435, 14666, 113366, 273141, 217015, 46656},
{1, 882, 48783, 600124, 2385999, 3487218, 1845697, 279936},
{1, 1777, 156933, 2937109, 17931235, 42397299, 40817623, 15159367, 1679616},
{1, 3568, 493900, 13631632, 121964374, 433696144, 667299052, 447815920, 121232113,10077696}
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MATHEMATICA
| M[p_, k_, x_] = ((-1)^k*(1 - x)^(p + k)/(k!(p - 1)!))*Sum[(p - 1 + j)!*j^k*x^j/(j!), {j, 0, Infinity}]/x;
Table[Table[CoefficientList[FullSimplify[ExpandAll[(-1)^m*m!*M[n, m, x]]]/n, x], {m, 1, 10}], {n, 1, 10}];
Table[Flatten[Table[CoefficientList[FullSimplify[ExpandAll[(-1)^m*m!*M[n, m, x]]]/n, x], {m, 1, 10}]], {n, 1, 10}]
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CROSSREFS
| A008292, A144696, A144697, A144698, A144699
Sequence in context: A157396 A019430 A064083 * A167580 A080213 A200091
Adjacent sequences: A152246 A152247 A152248 * A152250 A152251 A152252
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KEYWORD
| nonn
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 30 2008
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