This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227209 #45 Feb 23 2025 11:20:27
%S A227209 1,8,43,198,849,3516,14311,57746,231997,930024,3724179,14904894,
%T A227209 59635945,238576532,954371647,3817617642,15270732693,61083455040,
%U A227209 244334868715,977341571990,3909370482241,15637490317548
%N A227209 Expansion of 1/((1-x)^2*(1-2*x)*(1-4*x)).
%C A227209 This sequence was chosen to illustrate a method of solution.
%C A227209 In general, for the expansion of 1/((1-t*x)^2*(1-s*x)*(1-r*x)) with r>s>t we have the formula: a(n) = ( K*r^(n+3) + L*s^(n+3) + M*t^(n+3) + N*t^(n+3) )/D where K,L,M,N,D have the following values:
%C A227209 K = (s-t)^2;
%C A227209 L = -(r-t)^2;
%C A227209 M = (r-s)*(r+s-2*t);
%C A227209 N = (r-t)*(s-t)*(r-s)*(n+3);
%C A227209 D = (r-s)*(r-t)^2*(s-t)^2.
%C A227209 Directly using formula we get a(n) = ( 4^(n+3) - 9*2^(n+3) + 8 + 6*(n+3) )/18. After transformation we obtain previous formula.
%H A227209 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,22,-8).
%F A227209 G.f.: 1/((1-x)^2*(1-2*x)*(1-4*x)).
%F A227209 a(n) = ( 4^(n+3) - 9*2^(n+3) + 6*n + 26 )/18.
%F A227209 E.g.f.: exp(x)*(13 - 36*exp(x) + 32*exp(3*x) + 3*x)/9. - _Stefano Spezia_, Feb 23 2025
%t A227209 nn = 25; CoefficientList[Series[1/((1 - x)^2*(1 - 2 x)*(1 - 4 x)), {x, 0, nn}], x] (* _T. D. Noe_, Sep 19 2013 *)
%Y A227209 Cf. A229026.
%Y A227209 Partial sums of A171477.
%K A227209 nonn,easy
%O A227209 0,2
%A A227209 _Yahia Kahloune_, Sep 19 2013