# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a166346 Showing 1-1 of 1 %I A166346 #10 Dec 18 2022 07:06:55 %S A166346 1,1,1,1,8,1,1,39,39,1,1,158,482,158,1,1,605,4194,4194,605,1,1,2276, %T A166346 31047,67752,31047,2276,1,1,8515,210609,856075,856075,210609,8515,1,1, %U A166346 31802,1356368,9367974,17194910,9367974,1356368,31802,1,1,118713 %N 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). %D A166346 Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91. %F A166346 p(x,0)= 1/(1 - x); %F A166346 p(x,1)= x/(1 - x)^2; %F A166346 p(x,2)= x*(1 + x)/(1 - x)^3; %F A166346 p(x,3)= x*(x^2 +8*x + 1)/(1 - x)^4; %F A166346 p(x,n)= 2*x*D[p[x, n - 1], x] - p[x, n - 2] %e A166346 {1}, %e A166346 {1, 1}, %e A166346 {1, 8, 1}, %e A166346 {1, 39, 39, 1}, %e A166346 {1, 158, 482, 158, 1}, %e A166346 {1, 605, 4194, 4194, 605, 1}, %e A166346 {1, 2276, 31047, 67752, 31047, 2276, 1}, %e A166346 {1, 8515, 210609, 856075, 856075, 210609, 8515, 1}, %e A166346 {1, 31802, 1356368, 9367974, 17194910, 9367974, 1356368, 31802, 1}, %e A166346 {1, 118713, 8453460, 93489572, 285010254, 285010254, 93489572, 8453460, 118713, 1}, %e A166346 {1, 443072, 51564829, 876484896, 4159141218, 6855899968, 4159141218, 876484896, 51564829, 443072, 1} %t A166346 p[x_, 0] := 1/(1 - x); %t A166346 p[x_, 1] := x/(1 - x)^2; %t A166346 p[x_, 2] := x*(1 + x)/(1 - x)^3; %t A166346 p[x_, 3] := x*(x^2 + 8*x + 1)/(1 - x)^4; %t A166346 p[x_, n_] := p[x, n] = 2*x*D[p[x, n - 1], x] - p[x, n - 2] %t A166346 a = Table[CoefficientList[FullSimplify[ExpandAll[(1 - x)^(n + 1)*p[x, n]/x]], x], {n, 1, 11}]; %t A166346 Flatten[a] %t A166346 Table[Apply[Plus, CoefficientList[FullSimplify[ExpandAll[(1 - x)^(n + 1)*p[x, n]/x]], x]], {n, 1, 11}]; %Y A166346 Cf. A123125, A142458. %K A166346 nonn,tabl,uned,less %O A166346 1,5 %A A166346 _Roger L. Bagula_, Oct 12 2009 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE