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A290992 p-INVERT of (0,0,0,1,2,3,4,5,...), the nonnegative integers A000027 preceded by two zeros, where p(S) = 1 - S - S^2.

%I A290992 #13 Apr 12 2023 08:06:46
%S A290992 0,0,0,1,2,3,4,7,14,27,48,82,140,242,420,726,1250,2153,3720,6446,
%T A290992 11184,19408,33676,58431,101378,175861,304988,528800,916714,1589091,
%U A290992 2754612,4775074,8277754,14350253,24878304,43131381,74777890,129645147,224770632
%N A290992 p-INVERT of (0,0,0,1,2,3,4,5,...), the nonnegative integers A000027 preceded by two zeros, where p(S) = 1 - S - S^2.
%C A290992 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 A290992 See A290890 for a guide to related sequences.
%H A290992 G. C. Greubel, <a href="/A290992/b290992.txt">Table of n, a(n) for n = 0..1000</a>
%H A290992 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,0,-2,1,0,1).
%F A290992 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 2*a(n-5) + a(n-6) + a(n-8).
%F A290992 G.f.: x^3*(1 - 2*x + x^2 + x^4) / (1 - 4*x + 6*x^2 - 4*x^3 + 2*x^5 - x^6 - x^8). - _Colin Barker_, Aug 24 2017
%t A290992 z = 60; s = x^4/(1 - x)^2; p = 1 - s - s^2;
%t A290992 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* 0,0,0,1,2,3,4,5,... *)
%t A290992 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290992 *)
%o A290992 (PARI) concat(vector(3), Vec(x^3*(1 - 2*x + x^2 + x^4) / (1 - 4*x + 6*x^2 - 4*x^3 + 2*x^5 - x^6 - x^8) + O(x^50))) \\ _Colin Barker_, Aug 24 2017
%o A290992 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0,0] cat Coefficients(R!( x^3*(1-2*x+x^2+x^4)/(1-4*x+6*x^2-4*x^3+2*x^5-x^6-x^8) )); // _G. C. Greubel_, Apr 12 2023
%o A290992 (SageMath)
%o A290992 def f(x): return x^3*(1-2*x+x^2+x^4)/(1-4*x+6*x^2-4*x^3+2*x^5-x^6-x^8)
%o A290992 def A290992_list(prec):
%o A290992     P.<x> = PowerSeriesRing(ZZ, prec)
%o A290992     return P( f(x) ).list()
%o A290992 A290992_list(60) # _G. C. Greubel_, Apr 12 2023
%Y A290992 Cf. A000027, A033453, A289780, A290890, A290990, A290991.
%K A290992 nonn,easy
%O A290992 0,5
%A A290992 _Clark Kimberling_, Aug 21 2017