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A290997 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^3 - S^6.

%I A290997 #10 Apr 15 2023 02:04:14
%S A290997 0,0,1,3,6,12,27,63,143,315,684,1479,3195,6903,14932,32361,70266,
%T A290997 152775,332397,723330,1573829,3423444,7444722,16185939,35185779,
%U A290997 76483890,166253545,361396431,785621808,1707884880,3712912632,8071922817,17548551692,38150905170
%N A290997 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^3 - S^6.
%C A290997 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 A290997 See A291000 for a guide to related sequences.
%H A290997 Clark Kimberling, <a href="/A290997/b290997.txt">Table of n, a(n) for n = 0..1000</a>
%H A290997 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,21,-18,9,-1).
%F A290997 a(n) = 6*a(n-1) - 15*a(n-2) + 21*a(n-3) - 18*a(n-4) + 9*a(n-5) - a(n-6) for n >= 7.
%F A290997 G.f.: x^2*(1 - 3*x + 3*x^2) / (1 - 6*x + 15*x^2 - 21*x^3 + 18*x^4 - 9*x^5 + x^6). - _Colin Barker_, Aug 22 2017
%t A290997 z = 60; s = x/(1-x); p= 1 -s^3 -s^6;
%t A290997 Drop[CoefficientList[Series[s, {x,0,z}], x], 1]  (* A000012 *)
%t A290997 Drop[CoefficientList[Series[1/p, {x,0,z}], x], 1]  (* A290997 *)
%t A290997 LinearRecurrence[{6,-15,21,-18,9,-1}, {0,0,1,3,6,12}, 40] (* _G. C. Greubel_, Apr 14 2023 *)
%o A290997 (PARI) concat(vector(2), Vec(x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) + O(x^50))) \\ _Colin Barker_, Aug 22 2017
%o A290997 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) )); // _G. C. Greubel_, Apr 14 2023
%o A290997 (SageMath)
%o A290997 def A290997_list(prec):
%o A290997     P.<x> = PowerSeriesRing(ZZ, prec)
%o A290997     return P( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) ).list()
%o A290997 A290997_list(40) # _G. C. Greubel_, Apr 14 2023
%Y A290997 Cf. A000012, A033453, A289780, A291000.
%K A290997 nonn,easy
%O A290997 0,4
%A A290997 _Clark Kimberling_, Aug 22 2017