A166346 Coefficients of numerator of recursively defined rational function: p(x,3)=x*(x^2 + 8*x + 1)/(1 - x)^4; p(x, n) = 2*x*D[p(x, n - 1), x] - p(x,n-2).

1, 1, 1, 1, 8, 1, 1, 39, 39, 1, 1, 158, 482, 158, 1, 1, 605, 4194, 4194, 605, 1, 1, 2276, 31047, 67752, 31047, 2276, 1, 1, 8515, 210609, 856075, 856075, 210609, 8515, 1, 1, 31802, 1356368, 9367974, 17194910, 9367974, 1356368, 31802, 1, 1, 118713

Offset: 1

Author

Roger L. Bagula, Oct 12 2009

Keywords

• nonn
• tabl
• uned
• less

Examples

			{1},
{1, 1},
{1, 8, 1},
{1, 39, 39, 1},
{1, 158, 482, 158, 1},
{1, 605, 4194, 4194, 605, 1},
{1, 2276, 31047, 67752, 31047, 2276, 1},
{1, 8515, 210609, 856075, 856075, 210609, 8515, 1},
{1, 31802, 1356368, 9367974, 17194910, 9367974, 1356368, 31802, 1},
{1, 118713, 8453460, 93489572, 285010254, 285010254, 93489572, 8453460, 118713, 1},
{1, 443072, 51564829, 876484896, 4159141218, 6855899968, 4159141218, 876484896, 51564829, 443072, 1}
		

References

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

Crossrefs

Cf. A123125, A142458.

Programs

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 + 8*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}];

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 +8*x + 1)/(1 - x)^4;

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