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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). 5

%I #10 Dec 18 2022 09:27:04

%S 1,1,1,1,6,1,1,31,31,1,1,128,382,128,1,1,493,3346,3346,493,1,1,1858,

%T 24879,54044,24879,1858,1,1,6955,169209,683995,683995,169209,6955,1,1,

%U 25980,1091460,7496324,13738230,7496324,1091460,25980,1,1,96985,6809140

%N 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).

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

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

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

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

%F p(x,3)= x*(x^2 +6*x + 1)/(1 - x)^4;

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

%e {1},

%e {1, 1},

%e {1, 6, 1},

%e {1, 31, 31, 1},

%e {1, 128, 382, 128, 1},

%e {1, 493, 3346, 3346, 493, 1},

%e {1, 1858, 24879, 54044, 24879, 1858, 1},

%e {1, 6955, 169209, 683995, 683995, 169209, 6955, 1},

%e {1, 25980, 1091460, 7496324, 13738230, 7496324, 1091460, 25980, 1},

%e {1, 96985, 6809140, 74898500, 227852974, 227852974, 74898500, 6809140, 96985, 1},

%e {1, 361982, 41561069, 702794856, 3327271698, 5480955188, 3327271698, 702794856, 41561069, 361982, 1}

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

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

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

%t p[x_, 3] := x*(x^2 + 6*x + 1)/(1 - x)^4;

%t p[x_, n_] := p[x, n] = 2*x*D[p[x, n - 1], x] - p[x, n - 2]

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

%t Flatten[a]

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

%Y Cf. A123125.

%K nonn,tabl,uned,less

%O 1,5

%A _Roger L. Bagula_, Oct 12 2009

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Last modified July 15 00:37 EDT 2024. Contains 374323 sequences. (Running on oeis4.)