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.

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

Offset: 1

Author

Roger L. Bagula, Nov 30 2008

Keywords

• nonn
• tabl

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.

Examples

			{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}
		

References

• Douglas C. Montgomery, Lynwood A, Johnson, Forecasting and Time Series Analysis,McGraw-Hill, New York,1976,page 64.

Crossrefs

A008292, A144696, A144697, A144698, A144699

Programs

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}]

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