A166349 Coefficients of numerator of recursively defined rational function: p(x,3)=x*(x^2 + 6*x + 1)/(1 - x)^4; p(x, n) = 2*x*D[p(x, n - 1), x] - p(x,n-2).

1, 1, 1, 1, 6, 1, 1, 31, 31, 1, 1, 128, 382, 128, 1, 1, 493, 3346, 3346, 493, 1, 1, 1858, 24879, 54044, 24879, 1858, 1, 1, 6955, 169209, 683995, 683995, 169209, 6955, 1, 1, 25980, 1091460, 7496324, 13738230, 7496324, 1091460, 25980, 1, 1, 96985, 6809140

Offset

1,5

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..48.

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 +6*x + 1)/(1 - x)^4;

p(x,n)= 2*x*D[p[x, n - 1], x] - p[x, n - 2]

Example

{1},
{1, 1},
{1, 6, 1},
{1, 31, 31, 1},
{1, 128, 382, 128, 1},
{1, 493, 3346, 3346, 493, 1},
{1, 1858, 24879, 54044, 24879, 1858, 1},
{1, 6955, 169209, 683995, 683995, 169209, 6955, 1},
{1, 25980, 1091460, 7496324, 13738230, 7496324, 1091460, 25980, 1},
{1, 96985, 6809140, 74898500, 227852974, 227852974, 74898500, 6809140, 96985, 1},
{1, 361982, 41561069, 702794856, 3327271698, 5480955188, 3327271698, 702794856, 41561069, 361982, 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 + 6*x + 1)/(1 - x)^4;
p[x_, n_] := p[x, n] = 2*x*D[p[x, n - 1], x] - p[x, n - 2]
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

Cf. A123125.

Sequence in context: A265603 A174186 A111578 * A176429 A157155 A022169

Adjacent sequences: A166346 A166347 A166348 * A166350 A166351 A166352

Keyword

nonn,tabl,uned,less

Author

Roger L. Bagula, Oct 12 2009

Status

approved