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A291006 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3 - S^4.

%I A291006 #7 Jun 02 2023 21:50:15
%S A291006 1,3,9,27,80,235,688,2013,5891,17244,50482,147791,432672,1266680,
%T A291006 3708288,10856241,31782309,93044665,272394011,797450348,2334585333,
%U A291006 6834643282,20008841823,58577124509,171488162320,502042223184,1469759722591,4302812676894
%N A291006 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3 - S^4.
%C A291006 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 A291006 See A291000 for a guide to related sequences.
%H A291006 Clark Kimberling, <a href="/A291006/b291006.txt">Table of n, a(n) for n = 0..999</a>
%H A291006 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,6,-1).
%F A291006 G.f.: (1 - 2*x + 2*x^2)/(1 - 5*x + 8*x^2 - 6*x^3 + x^4).
%F A291006 a(n) = 5*a(n-1) - 8*a(n-2) + 6*a(n-3) - a(n-4) for n >= 4.
%t A291006 z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3 - s^4;
%t A291006 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291006 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291006 *)
%t A291006 LinearRecurrence[{5,-8,6,-1}, {1,3,9,27}, 41] (* _G. C. Greubel_, Jun 01 2023 *)
%o A291006 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-2*x+2*x^2)/(1-5*x+8*x^2-6*x^3+x^4) )); // _G. C. Greubel_, Jun 01 2023
%o A291006 (SageMath)
%o A291006 def A291006_list(prec):
%o A291006     P.<x> = PowerSeriesRing(ZZ, prec)
%o A291006     return P( (1-2*x+2*x^2)/(1-5*x+8*x^2-6*x^3+x^4) ).list()
%o A291006 A291006_list(40) # _G. C. Greubel_, Jun 01 2023
%Y A291006 Cf. A000012, A289780, A291000.
%K A291006 nonn,easy
%O A291006 0,2
%A A291006 _Clark Kimberling_, Aug 23 2017