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A291411 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2 + S^3.

%I A291411 #7 Sep 27 2017 09:24:59
%S A291411 2,7,21,63,189,567,1699,5092,15260,45731,137046,410697,1230768,
%T A291411 3688339,11053134,33123790,99264648,297474121,891463923,2671519536,
%U A291411 8005951162,23992058879,71898875923,215464974683,645700711159,1935021731510,5798830691535,17377808652745
%N A291411 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2 + S^3.
%C A291411 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 A291411 See A291382 for a guide to related sequences.
%H A291411 Clark Kimberling, <a href="/A291411/b291411.txt">Table of n, a(n) for n = 0..1000</a>
%H A291411 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 3, 1, -2, -3, -1)
%F A291411 G.f.: -(((-1 + x) (1 + x) (2 + x) (1 + x + x^2))/(1 - 2 x - 3 x^2 - x^3 + 2 x^4 + 3 x^5 + x^6)).
%F A291411 a(n) = 2*a(n-1) + 3*a(n-2) + a(n-3) - 2*a(n-4) - 3*a(n-5) - a(n-6) for n >= 7.
%t A291411 z = 60; s = x + x^2; p = 1 - 2 s - s^2 + s^3;
%t A291411 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
%t A291411 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291411 *)
%o A291411 (GAP)
%o A291411 a:=[2,7,21,63,189,567];;  for n in [7..10^2] do a[n]:=2*a[n-1]+3*a[n-2]+a[n-3]-2*a[n-4]-3*a[n-5]-a[n-6]; od; a; # _Muniru A Asiru_, Sep 12 2017
%Y A291411 Cf. A019590, A291382.
%K A291411 nonn,easy
%O A291411 0,1
%A A291411 _Clark Kimberling_, Sep 07 2017