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A291009 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S)*(1 - 3*S).

%I A291009 #11 Jun 04 2023 19:56:08
%S A291009 4,17,70,284,1144,4592,18400,73664,294784,1179392,4718080,18873344,
%T A291009 75495424,301985792,1207951360,4831821824,19327320064,77309345792,
%U A291009 309237514240,1236950319104,4947801800704,19791208251392,79164835102720,316659344605184,1266637386809344
%N A291009 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S)*(1 - 3*S).
%C A291009 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 A291009 See A291000 for a guide to related sequences.
%H A291009 Clark Kimberling, <a href="/A291009/b291009.txt">Table of n, a(n) for n = 0..1000</a>
%H A291009 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%F A291009 a(n) = 6*a(n-1) - 8*a(n-2) for n >= 3.
%F A291009 From _Colin Barker_, Aug 23 2017: (Start)
%F A291009 G.f.: (4 - 7*x) / ((1 - 2*x)*(1 - 4*x)).
%F A291009 a(n) = 2^(n-1) * (9*2^n - 1).
%F A291009 (End)
%t A291009 z = 60; s = x/(1-x); p = (1-s)(1-3s);
%t A291009 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291009 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291009 *)
%t A291009 LinearRecurrence[{6,-8}, {4,17}, 40] (* _G. C. Greubel_, Jun 04 2023 *)
%o A291009 (PARI) Vec((4-7*x)/((1-2*x)*(1-4*x)) + O(x^30)) \\ _Colin Barker_, Aug 23 2017
%o A291009 (Magma) [2^(n-1)*(9*2^n -1): n in [0..40]]; // _G. C. Greubel_, Jun 04 2023
%o A291009 (SageMath)
%o A291009 A291009=BinaryRecurrenceSequence(6,-8,4,17)
%o A291009 [A291009(n) for n in range(41)] # _G. C. Greubel_, Jun 04 2023
%Y A291009 Cf. A000012, A033453, A289780, A291000.
%K A291009 nonn,easy
%O A291009 0,1
%A A291009 _Clark Kimberling_, Aug 23 2017