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A289803 p-INVERT of the even bisection (A001906) of the Fibonacci numbers, where p(S) = 1 - S - S^2.

%I A289803 #14 May 03 2024 17:11:50
%S A289803 1,5,23,103,456,2009,8833,38803,170399,748176,3284833,14421533,
%T A289803 63314735,277968871,1220356440,5357681369,23521603225,103265890987,
%U A289803 453363808127,1990383615264,8738295434881,38363361811637,168425013526727,739429075564711,3246283590352104
%N A289803 p-INVERT of the even bisection (A001906) of the Fibonacci numbers, where p(S) = 1 - S - S^2.
%C A289803 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 A289803 See A289780 for a guide to related sequences.
%H A289803 Clark Kimberling, <a href="/A289803/b289803.txt">Table of n, a(n) for n = 0..1000</a>
%H A289803 Rigoberto Flórez, Javier González, Mateo Matijasevick, Cristhian Pardo, José Luis Ramírez, Lina Simbaqueba, and Fabio Velandia, <a href="https://cdm.ucalgary.ca/article/view/73812">Lattice paths in corridors and cyclic corridors</a>, Contrib. Disc. Math. (2024) Vol. 19. No. 2, 36-55. See p. 11.
%H A289803 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-13,7,-1).
%F A289803 G.f.: (1 - 2 x + x^2)/(1 - 7 x + 13 x^2 - 7 x^3 + x^4).
%F A289803 a(n) = 7*a(n-1) - 13*a(n-2) + 7*a(n-3) - a(n-4).
%t A289803 z = 60; s = x/(1 - 3*x + x^2); p = 1 - s - s^2;
%t A289803 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A001906 *)
%t A289803 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289803 *)
%Y A289803 Cf. A001906, A289780, A298804.
%K A289803 nonn,easy
%O A289803 0,2
%A A289803 _Clark Kimberling_, Aug 12 2017