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A292322 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = 1 - S - S^2 - S^3.

%I A292322 #4 Sep 23 2017 19:21:30
%S A292322 1,2,4,8,17,36,73,152,317,653,1355,2812,5818,12061,25001,51786,107323,
%T A292322 222409,460824,954942,1978840,4100398,8496827,17606974,36484494,
%U A292322 75602461,156661630,324629762,672690133,1393931744,2888469094,5985414154,12402824741
%N A292322 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = 1 - S - S^2 - S^3.
%C A292322 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 A292322 See A291728 for a guide to related sequences.
%H A292322 Clark Kimberling, <a href="/A292322/b292322.txt">Table of n, a(n) for n = 0..1000</a>
%H A292322 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 4, -2, -1, -3, 1, 0, 1)
%F A292322 G.f.: -((1 + x + x^2 - 2 x^3 - x^4 + x^6)/(-1 + x + x^2 + 4 x^3 - 2 x^4 -  x^5 - 3 x^6 + x^7 + x^9)).
%F A292322 a(n) = a(n-1) + a(n-2) + 4*a(n-3) - 2*a(n-4) - a(n-5) - 3*a(n-6)  + a(n-7) + a(n-9) for n >= 10.
%t A292322 z = 60; s = x/(x - x^3); p = 1 - s - s^2 - s^3;
%t A292322 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A079978  *)
%t A292322 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292322 *)
%Y A292322 Cf. A079978, A292320.
%K A292322 nonn,easy
%O A292322 0,2
%A A292322 _Clark Kimberling_, Sep 15 2017