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A289782 p-INVERT of the Lucas numbers (A000032), where p(S) = 1 - S - S^2.

%I A289782 #22 Aug 19 2017 13:32:07
%S A289782 2,9,35,146,593,2428,9911,40495,165399,675637,2759792,11273144,
%T A289782 46048100,188095781,768327108,3138436438,12819777601,52365789305,
%U A289782 213901984464,873739509697,3569021260182,14578615958179,59550231769665,243248749683441,993614171826023
%N A289782 p-INVERT of the Lucas numbers (A000032), where p(S) = 1 - S - S^2.
%C A289782 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 A289782 See A289780 for a guide to related sequences.
%H A289782 Clark Kimberling, <a href="/A289782/b289782.txt">Table of n, a(n) for n = 0..1000</a>
%H A289782 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, 2, -7, 1)
%F A289782 G.f.: (-2 - x + 5 x^2 - 2 x^3)/(-1 + 4 x + 2 x^2 - 7 x^3 + x^4).
%F A289782 a(n) = 4*a(n-1) + 2*a(n-2) - 7*a(n-3) + a(n-4).
%t A289782 z = 60; s = (2 - x) x/(1 - x - x^2); p = 1 - s - s^2;
%t A289782 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000032 *)
%t A289782 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289782 *)
%Y A289782 Cf. A000032, A289780.
%K A289782 nonn,easy
%O A289782 0,1
%A A289782 _Clark Kimberling_, Aug 10 2017