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A291011 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S)^2 * (1 - 2*S).

%I A291011 #14 Jun 05 2023 07:37:13
%S A291011 4,15,52,172,552,1736,5384,16536,50440,153112,463176,1397720,4210568,
%T A291011 12668568,38083528,114414424,343587336,1031482904,3095956040,
%U A291011 9291013848,27879595144,83652416920,250985562312,753015407192,2259167856392,6777755227416,20333785775944
%N A291011 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S)^2 * (1 - 2*S).
%C A291011 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 A291011 See A291000 for a guide to related sequences.
%H A291011 Clark Kimberling, <a href="/A291011/b291011.txt">Table of n, a(n) for n = 0..999</a>
%H A291011 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-16,12).
%F A291011 G.f.: (4 - 13*x + 11*x^2)/((1-2*x)^2 * (1-3*x)).
%F A291011 a(n) = 7*a(n-1) - 16*a(n-2) + 12*a(n-3) for n >= 4.
%F A291011 a(n) = 8*3^n - 2^(n-1)*(8+n). - _Colin Barker_, Aug 23 2017
%F A291011 E.g.f.: 8*exp(3*x) - (4 + x)*exp(2*x). - _G. C. Greubel_, Jun 04 2023
%t A291011 z = 60; s = x/(1-x); p = (1-s)^2*(1-2*s);
%t A291011 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291011 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* this sequence *)
%t A291011 LinearRecurrence[{7,-16,12},{4,15,52},30] (* _Harvey P. Dale_, Sep 23 2017 *)
%o A291011 (PARI) Vec((4 -13*x +11*x^2)/((1-2*x)^2*(1-3*x)) + O(x^30)) \\ _Colin Barker_, Aug 23 2017
%o A291011 (Magma) [8*3^n - 2^(n-1)*(8+n): n in [0..40]]; // _G. C. Greubel_, Jun 04 2023
%o A291011 (SageMath) [8*3^n - 2^(n-1)*(8+n) for n in range(41)] # _G. C. Greubel_, Jun 04 2023
%Y A291011 Cf. A000012, A033453, A289780, A291000.
%K A291011 nonn,easy
%O A291011 0,1
%A A291011 _Clark Kimberling_, Aug 23 2017