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A291010 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - 2*S)*(1 - 3*S).

%I A291010 #14 Jun 05 2023 07:21:20
%S A291010 5,24,108,468,1980,8244,33948,138708,563580,2280564,9200988,37040148,
%T A291010 148869180,597602484,2396787228,9606280788,38482518780,154102262004,
%U A291010 616925608668,2469252116628,9881657512380,39540577187124,158204150161308,632942124883668
%N A291010 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - 2*S)*(1 - 3*S).
%C A291010 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 A291010 See A291000 for a guide to related sequences.
%H A291010 Clark Kimberling, <a href="/A291010/b291010.txt">Table of n, a(n) for n = 0..1000</a>
%H A291010 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-12).
%F A291010 G.f.: (5 - 11*x)/(1 - 7*x + 12*x^2).
%F A291010 a(n) = 7*a(n-1) - 12*a(n-2) for n >= 3.
%F A291010 a(n) = 9*4^n - 4*3^n. - _Colin Barker_, Aug 23 2017
%F A291010 E.g.f.: 9*exp(4*x) - 4*exp(3*x). - _G. C. Greubel_, Jun 04 2023
%t A291010 z = 60; s = x/(1-x); p = (1-2s)(1-3s);
%t A291010 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291010 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* this sequence *)
%t A291010 LinearRecurrence[{7,-12}, {5,24}, 40] (* _G. C. Greubel_, Jun 04 2023 *)
%o A291010 (PARI) Vec((5-11*x)/((1-3*x)*(1-4*x)) + O(x^30)) \\ _Colin Barker_, Aug 23 2017
%o A291010 (Magma) [36*(4^(n-1)-3^(n-2)): n in [0..40]]; // _G. C. Greubel_, Jun 04 2023
%o A291010 (SageMath)
%o A291010 A291010=BinaryRecurrenceSequence(7,-12,5,24)
%o A291010 [A291010(n) for n in range(41)] # _G. C. Greubel_, Jun 04 2023
%Y A291010 Cf. A000012, A033453, A289780, A291000.
%K A291010 nonn,easy
%O A291010 0,1
%A A291010 _Clark Kimberling_, Aug 23 2017