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A291245 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - 5 S + S^2.

%I A291245 #9 Aug 29 2017 03:30:14
%S A291245 5,24,120,599,2990,14925,74500,371876,1856265,9265776,46251265,
%T A291245 230868900,1152410620,5752399899,28713814350,143328549649,
%U A291245 715442152480,3571217840400,17826174791885,88981552487776,444162405876285,2217091490069376,11066885918992400
%N A291245 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - 5 S + S^2.
%C A291245 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 A291245 See A291219 for a guide to related sequences.
%H A291245 Clark Kimberling, <a href="/A291245/b291245.txt">Table of n, a(n) for n = 0..1000</a>
%H A291245 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,1,-5,-1)
%F A291245 G.f.: (5 - x - 5*x^2)/(1 - 5*x - x^2 + 5*x^3 + x^4).
%F A291245 a(n) = 5*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) for n >= 5.
%t A291245 z = 60; s = x/(1 - x^2); p = 1 - 5 s - s^2;
%t A291245 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000035 *)
%t A291245 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291245 *)
%Y A291245 Cf. A000035, A291219.
%K A291245 nonn,easy
%O A291245 0,1
%A A291245 _Clark Kimberling_, Aug 28 2017