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%I #7 Jan 09 2024 12:23:09
%S 1,1,6,1,19,36,1,46,241,216,1,101,1091,2551,1296,1,212,4182,18932,
%T 24337,7776,1,435,14666,113366,273141,217015,46656,1,882,48783,600124,
%U 2385999,3487218,1845697,279936,1,1777,156933,2937109,17931235,42397299
%N 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
%C 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.
%D Douglas C. Montgomery, Lynwood A, Johnson, Forecasting and Time Series Analysis,McGraw-Hill, New York,1976,page 64.
%F 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;
%F t(m,l)=coefficients((-1)^m*m!*M[n, m, x])/n
%e {1},
%e {1, 6},
%e {1, 19, 36},
%e {1, 46, 241, 216},
%e {1, 101, 1091, 2551, 1296},
%e {1, 212, 4182, 18932, 24337, 7776},
%e {1, 435, 14666, 113366, 273141, 217015, 46656},
%e {1, 882, 48783, 600124, 2385999, 3487218, 1845697, 279936},
%e {1, 1777, 156933, 2937109, 17931235, 42397299, 40817623, 15159367, 1679616},
%e {1, 3568, 493900, 13631632, 121964374, 433696144, 667299052, 447815920, 121232113,10077696}
%t 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;
%t Table[Table[CoefficientList[FullSimplify[ExpandAll[(-1)^m*m!*M[n, m, x]]]/n, x], {m, 1, 10}], {n, 1, 10}];
%t Table[Flatten[Table[CoefficientList[FullSimplify[ExpandAll[(-1)^m*m!*M[n, m, x]]]/n, x], {m, 1, 10}]], {n, 1, 10}]
%Y A008292, A144696, A144697, A144698, A144699
%K nonn,tabl
%O 1,3
%A _Roger L. Bagula_, Nov 30 2008
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