A166346 - OEIS

%I #10 Dec 18 2022 07:06:55

%S 1,1,1,1,8,1,1,39,39,1,1,158,482,158,1,1,605,4194,4194,605,1,1,2276,

%T 31047,67752,31047,2276,1,1,8515,210609,856075,856075,210609,8515,1,1,

%U 31802,1356368,9367974,17194910,9367974,1356368,31802,1,1,118713

%N 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).

%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 +8*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, 8, 1},

%e {1, 39, 39, 1},

%e {1, 158, 482, 158, 1},

%e {1, 605, 4194, 4194, 605, 1},

%e {1, 2276, 31047, 67752, 31047, 2276, 1},

%e {1, 8515, 210609, 856075, 856075, 210609, 8515, 1},

%e {1, 31802, 1356368, 9367974, 17194910, 9367974, 1356368, 31802, 1},

%e {1, 118713, 8453460, 93489572, 285010254, 285010254, 93489572, 8453460, 118713, 1},

%e {1, 443072, 51564829, 876484896, 4159141218, 6855899968, 4159141218, 876484896, 51564829, 443072, 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 + 8*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, A142458.

%K nonn,tabl,uned,less

%O 1,5

%A _Roger L. Bagula_, Oct 12 2009