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A291400 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - S^2 - S^4.

%I A291400 #4 Sep 06 2017 21:16:21
%S A291400 0,1,2,3,8,15,26,52,100,193,378,726,1396,2699,5210,10065,19444,37528,
%T A291400 72448,139890,270104,521547,1007026,1944336,3754132,7248558,13995676,
%U A291400 27023186,52176848,100743849,194517966,375578833,725174524,1400180233,2703493026
%N A291400 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - S^2 - S^4.
%C A291400 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 A291400 See A291382 for a guide to related sequences.
%H A291400 Clark Kimberling, <a href="/A291400/b291400.txt">Table of n, a(n) for n = 0..1000</a>
%H A291400 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 2, 2, 4, 6, 4, 1)
%F A291400 G.f.: -((x (1 + x)^2 (1 + x^2 + 2 x^3 + x^4))/(-1 + x^2 + 2 x^3 + 2 x^4 +   4 x^5 + 6 x^6 + 4 x^7 + x^8)).
%F A291400 a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) + 4*a(n-5) + 6*a(n-6) + 4*a(n-7) + a(n-8) for n >= 9.
%t A291400 z = 60; s = x + x^2; p = 1 - s^2 - s^4;
%t A291400 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
%t A291400 u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291400 *)
%Y A291400 Cf. A019590, A291382.
%K A291400 nonn,easy
%O A291400 0,3
%A A291400 _Clark Kimberling_, Sep 06 2017