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A291002 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S)*(1 - 2*S)*(1 - 3*S).

%I A291002 #12 Apr 28 2023 08:16:11
%S A291002 6,31,146,652,2816,11896,49496,203752,832376,3381736,13683896,
%T A291002 55206952,222242936,893219176,3585623096,14380739752,57637717496,
%U A291002 230895178216,924613703096,3701553914152,14815513224056,59289946122856,237243465219896,949224905162152
%N A291002 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S)*(1 - 2*S)*(1 - 3*S).
%C A291002 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 A291002 See A291000 for a guide to related sequences.
%H A291002 Clark Kimberling, <a href="/A291002/b291002.txt">Table of n, a(n) for n = 0..1000</a>
%H A291002 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-26,24).
%F A291002 G.f.: (6 - 23*x + 23*x^2)/(1 - 9*x + 26*x^2 - 24*x^3).
%F A291002 a(n) = 9*a(n-1) - 26*a(n-2) + 24*a(n-3) for n >= 4.
%F A291002 a(n) = (2^n - 16*3^n + 27*4^n) / 2. - _Colin Barker_, Aug 23 2017
%F A291002 E.g.f.: (1/2)*(exp(2*x) - 16*exp(3*x) + 27*exp(4*x)). - _G. C. Greubel_, Apr 27 2023
%t A291002 z = 60; s = x/(1-x); p = (1-s)*(1-2*s)*(1-3*s);
%t A291002 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291002 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291002 *)
%t A291002 LinearRecurrence[{9,-26,24}, {6,31,146}, 41] (* _G. C. Greubel_, Apr 27 2023 *)
%o A291002 (Magma) [(2^n-16*3^n+27*4^n)/2: n in [0..40]]; // _G. C. Greubel_, Apr 27 2023
%o A291002 (SageMath) [(2^n-16*3^n+27*4^n)/2 for n in range(41)] # _G. C. Greubel_, Apr 27 2023
%Y A291002 Cf. A000012, A033453, A289780, A291000.
%K A291002 nonn,easy
%O A291002 0,1
%A A291002 _Clark Kimberling_, Aug 22 2017