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A291239 p-INVERT of (0,1,0,1,0,1,...), where p(S) = (1 - S^2) (1 - 2 S). 2

%I

%S 2,5,12,31,74,184,442,1081,2604,6323,15250,36912,89074,215293,519660,

%T 1255223,3030106,7317032,17664170,42649553,102963276,248587051,

%U 600137378,1448890464,3497918306,8444802101,20387522508,49220043535,118827609578,286875776920

%N p-INVERT of (0,1,0,1,0,1,...), where p(S) = (1 - S^2) (1 - 2 S).

%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 A291219 for a guide to related sequences.

%H Clark Kimberling, <a href="/A291239/b291239.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 4, -6, -4, 2, 1)

%F G.f.: (-2 - x + 6 x^2 + x^3 - 2 x^4)/(-1 + 2 x + 4 x^2 - 6 x^3 - 4 x^4 + 2 x^5 + x^6).

%F a(n) = 2*a(n-1) + 4*a(n-2) - 6*a(n-3) - 4*a(n-4) + 2*a(n-5) + a(n-6) for n >= 7.

%t z = 60; s = x/(1 - x^2); p = (1 - s^2)(1 - 2s);

%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291239 *)

%Y Cf. A000035, A291219.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Aug 28 2017

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Last modified August 1 14:40 EDT 2021. Contains 346391 sequences. (Running on oeis4.)