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A291239 p-INVERT of (0,1,0,1,0,1,...), where p(S) = (1 - S^2) (1 - 2 S).

%I A291239 #4 Aug 28 2017 20:15:25
%S A291239 2,5,12,31,74,184,442,1081,2604,6323,15250,36912,89074,215293,519660,
%T A291239 1255223,3030106,7317032,17664170,42649553,102963276,248587051,
%U A291239 600137378,1448890464,3497918306,8444802101,20387522508,49220043535,118827609578,286875776920
%N A291239 p-INVERT of (0,1,0,1,0,1,...), where p(S) = (1 - S^2) (1 - 2 S).
%C A291239 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 A291239 See A291219 for a guide to related sequences.
%H A291239 Clark Kimberling, <a href="/A291239/b291239.txt">Table of n, a(n) for n = 0..1000</a>
%H A291239 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 4, -6, -4, 2, 1)
%F A291239 G.f.: (-2 - x + 6 x^2 + x^3 - 2 x^4)/(-1 + 2 x + 4 x^2 - 6 x^3 - 4 x^4 + 2 x^5 + x^6).
%F A291239 a(n) = 2*a(n-1) + 4*a(n-2) - 6*a(n-3) - 4*a(n-4) + 2*a(n-5) + a(n-6) for n >= 7.
%t A291239 z = 60; s = x/(1 - x^2); p = (1 - s^2)(1 - 2s);
%t A291239 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000035 *)
%t A291239 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291239 *)
%Y A291239 Cf. A000035, A291219.
%K A291239 nonn,easy
%O A291239 0,1
%A A291239 _Clark Kimberling_, Aug 28 2017