|
Search: id:A152249
|
|
|
| A152249 |
|
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 |
|
+0
1
|
|
| 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; listen)
|
|
|
OFFSET
|
1,3
|
|
|
COMMENT
|
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.
|
|
REFERENCES
|
Douglas C. Montgomery, Lynwood A, Johnson, Forecasting and Time Series Analysis,McGraw-Hill, New York,1976,page 64.
|
|
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
|
|
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}
|
|
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}]
|
|
CROSSREFS
|
A008292, A144696, A144697, A144698, A144699
Sequence in context: A157396 A019430 A064083 this_sequence A167580 A080213 A161151
Adjacent sequences: A152246 A152247 A152248 this_sequence A152250 A152251 A152252
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 30 2008
|
|
|
Search completed in 0.002 seconds
|
