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A291253 p-INVERT of (0,1,0,1,0,1,...), where p(S) = (1 - S - S^2)^2.

%I A291253 #9 Sep 07 2017 14:28:17
%S A291253 2,5,12,30,70,166,382,881,2002,4540,10210,22891,51050,113506,251430,
%T A291253 555466,1223680,2689591,5898290,12909880,28204178,61515521,133961048,
%U A291253 291308806,632628710,1372170030,2972790738,6433570445,13909116418,30042364980,64830556978
%N A291253 p-INVERT of (0,1,0,1,0,1,...), where p(S) = (1 - S - S^2)^2.
%C A291253 Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
%C A291253 See A291219 for a guide to related sequences.
%H A291253 Clark Kimberling, <a href="/A291253/b291253.txt">Table of n, a(n) for n = 0..1000</a>
%H A291253 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2, 5, -8, -9, 8, 5, -2, -1)
%F A291253 G.f.: (2 + x - 8 x^2 - 3 x^3 + 8 x^4 + x^5 - 2 x^6)/(1 - x - 3 x^2 + x^3 + x^4)^2.
%F A291253 a(n) = 2*a(n-1) + 5*a(n-2) - 8*a(n-3) - 9*a(n-4) + 8*a(n-5) + 5*a(n-6) - 2*a(n-7) - a(n-8) for n >= 9.
%t A291253 z = 60; s = x/(1 - x^2); p = (1 - s - s^2)^2;
%t A291253 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000035 *)
%t A291253 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291253 *)
%o A291253 (GAP)
%o A291253 P:=[2,5,12,30,70,166,382,881];; for n in [9..10^3] do P[n]:=2*P[n-1]+5*P[n-2]-8*P[n-3]-9*P[n-4]+8*P[n-5]+5*P[n-6]-2*P[n-7]-P[n-8]; od; P;  # _Muniru A Asiru_, Sep 03 2017
%Y A291253 Cf. A000035, A291219.
%K A291253 nonn,easy
%O A291253 0,1
%A A291253 _Clark Kimberling_, Aug 31 2017