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A290928 p-INVERT of the positive integers, where p(S) = (1 - S^3)^2.

%I A290928 #15 Aug 20 2017 11:07:02
%S A290928 0,0,2,12,42,115,288,738,2020,5751,16362,45697,125538,342318,933050,
%T A290928 2547630,6960042,18990309,51699042,140439411,380871538,1031705466,
%U A290928 2792009100,7548723827,20389716864,55020917232,148334534420,399562167420,1075432476492
%N A290928 p-INVERT of the positive integers, where p(S) = (1 - S^3)^2.
%C A290928 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 A290928 See A290890 for a guide to related sequences.
%H A290928 Clark Kimberling, <a href="/A290928/b290928.txt">Table of n, a(n) for n = 0..1000</a>
%H A290928 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,222,-507,822,-965,822,-507,222,-66,12,-1)
%F A290928 a(n) = 12*a(n-1) - 66*a(n-2) + 222*a(n-3) - 507*a(n-4) + 822*a(n-5) - 965*a(n-6) + 822*a(n-7) - 507*a(n-8) + 222*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12).
%F A290928 G.f.: x^2*(2 - 12*x + 30*x^2 - 41*x^3 + 30*x^4 - 12*x^5 + 2*x^6) / ((1 - 3*x + x^2)^2*(1 - 3*x + 5*x^2 - 3*x^3 + x^4)^2). - _Colin Barker_, Aug 19 2017
%t A290928 z = 60; s = x/(1 - x)^2; p = (1 - s^3)^2;
%t A290928 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
%t A290928 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290928 *)
%t A290928 LinearRecurrence[{12, -66, 222, -507, 822, -965, 822, -507, 222, -66, 12, -1}, {0, 0, 2, 12, 42, 115, 288, 738, 2020, 5751, 16362, 45697}, 40] (* _Vincenzo Librandi_, Aug 20 2017 *)
%o A290928 (PARI) concat(vector(2), Vec(x^2*(2 - 12*x + 30*x^2 - 41*x^3 + 30*x^4 - 12*x^5 + 2*x^6) / ((1 - 3*x + x^2)^2*(1 - 3*x + 5*x^2 - 3*x^3 + x^4)^2) + O(x^30))) \\ _Colin Barker_, Aug 19 2017
%Y A290928 Cf. A000027, A033453, A290890.
%K A290928 nonn,easy
%O A290928 0,3
%A A290928 _Clark Kimberling_, Aug 19 2017