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%I #11 Sep 08 2022 08:46:19
%S 4,22,116,608,3180,16618,86812,453440,2368292,12369174,64601428,
%T 337397536,1762142540,9203221994,48066074172,251036784256,
%U 1311100720708,6847542588950,35762957380148,186780746599392,975507894703660,5094827328491242,26608975328086364
%N p-INVERT of the positive integers, where p(S) = 1 - 4*S + 2*S^2.
%C 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 See A290890 for a guide to related sequences.
%H Clark Kimberling, <a href="/A291183/b291183.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16,8,-1)
%F G.f.: (2 (2 - 5 x + 2 x^2))/(1 - 8 x + 16 x^2 - 8 x^3 + x^4).
%F a(n) = 8*a(n-1) - 16*a(n-2) + 8*a(n-3) - a(n-4).
%t z = 60; s = x/(1 - x)^2; p = 1 - 4 s + 2 s^2;
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291183 *)
%t LinearRecurrence[{8, -16, 8, -1}, {4, 22, 116, 608}, 40] (* _Vincenzo Librandi_, Aug 20 2017 *)
%o (Magma) I:=[4,22,116,608]; [n le 4 select I[n] else 8*Self(n-1)-16*Self(n-2)+8*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Aug 20 2017
%Y Cf. A000027, A290890.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Aug 19 2017
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