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A166345 Coefficients of recursive differential polynomial:p(x,3)=x*(x^2 + 2*x + 1)/(1 - x)^4;p(x, n) = x*D[p[x, n - 1], x] 0
1, 1, 1, 1, 2, 1, 1, 7, 7, 1, 1, 18, 42, 18, 1, 1, 41, 198, 198, 41, 1, 1, 88, 799, 1584, 799, 88, 1, 1, 183, 2925, 10331, 10331, 2925, 183, 1, 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1, 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1, 1, 1524 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums are:{1, 2, 4, 16, 80, 480, 3360, 26880, 241920, 2419200, 26611200,...}

REFERENCES

Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

LINKS

Table of n, a(n) for n=1..57.

FORMULA

p(x,0)= 1/(1 - x);

p(x,1)= x/(1 - x)^2;

p(x,2)= x*(1 + x)/(1 - x)^3;

p(x,3)= x*(x^2 + 2*x + 1)/(1 - x)^4;

p(x,n)= x*D[p[x, n - 1], x]

EXAMPLE

{1},

{1, 1},

{1, 2, 1},

{1, 7, 7, 1},

{1, 18, 42, 18, 1},

{1, 41, 198, 198, 41, 1},

{1, 88, 799, 1584, 799, 88, 1},

{1, 183, 2925, 10331, 10331, 2925, 183, 1},

{1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1},

{1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1},

{1, 1524, 106293, 1488832, 6490186, 10437528, 6490186, 1488832, 106293, 1524, 1}

MATHEMATICA

p[x_, 0] := 1/(1 - x);

p[x_, 1] := x/(1 - x)^2;

p[x_, 2] := x*(1 + x)/(1 - x)^3;

p[x_, 3] := x*(x^2 + 2*x + 1)/(1 - x)^4;

p[x_, n_] := p[x, n] = x*D[p[x, n - 1], x]

a = Table[CoefficientList[FullSimplify[ExpandAll[(1 - x)^(n + 1)*p[x, n]/x]], x], {n, 1, 11}];

Flatten[a]

Table[Apply[Plus, CoefficientList[FullSimplify[ExpandAll[(1 - x)^(n + 1)*p[x, n]/x]], x]], {n, 1, 11}];

CROSSREFS

A123125

Sequence in context: A303817 A158200 A220602 * A015110 A128596 A176305

Adjacent sequences:  A166342 A166343 A166344 * A166346 A166347 A166348

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Oct 12 2009

STATUS

approved

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Last modified May 13 05:02 EDT 2021. Contains 343836 sequences. (Running on oeis4.)