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A291143 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S^3)^3.

%I A291143 #13 Sep 01 2017 23:53:59
%S A291143 0,0,3,9,18,36,81,189,430,954,2097,4602,10080,21996,47796,103473,
%T A291143 223308,480584,1031571,2208807,4718610,10058580,21398715,45438270,
%U A291143 96313626,203812110,430615240,908455203,1913845374,4026531804,8460687861,17756508321,37223049942
%N A291143 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S^3)^3.
%C A291143 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 A291143 See A291000 for a guide to related sequences.
%H A291143 Clark Kimberling, <a href="/A291143/b291143.txt">Table of n, a(n) for n = 0..1000</a>
%H A291143 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,87,-144,171,-147,90,-36,8)
%F A291143 a(n) = 9*a(n-1) - 36 a(n-2) + 87*a(n-3) - 144*a(n-4) + 171*a(n-5) - 147*a(n-6) + 90*a(n-7) - 36*a(n-8) + 8*a(n-9) for n >= 10.
%F A291143 G.f.: x^2*(3 - 18*x + 45*x^2 - 63*x^3 + 54*x^4 - 27*x^5 + 7*x^6) / ((1 - 2*x)^3*(1 - x + x^2)^3). - _Colin Barker_, Aug 24 2017
%t A291143 z = 60; s = x/(1 - x); p = (1 - s^3)^3;
%t A291143 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291143 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291143 *)
%t A291143 LinearRecurrence[{9, -36, 87, -144, 171, -147, 90, -36, 8}, {0, 0, 3, 9, 18, 36, 81, 189, 430}, 40] (* _Vincenzo Librandi_, Aug 29 2017 *)
%o A291143 (PARI) concat(vector(2), Vec(x^2*(3 - 18*x + 45*x^2 - 63*x^3 + 54*x^4 - 27*x^5 + 7*x^6) / ((1 - 2*x)^3*(1 - x + x^2)^3) + O(x^30))) \\ _Colin Barker_, Aug 24 2017
%Y A291143 Cf. A000012, A289780, A291000.
%K A291143 nonn,easy
%O A291143 0,3
%A A291143 _Clark Kimberling_, Aug 24 2017