A292320 - OEIS

%I #5 Sep 14 2017 18:15:22

%S 1,1,2,4,6,12,22,36,67,122,209,377,681,1193,2130,3823,6764,12043,

%T 21531,38252,68076,121456,216126,384691,685636,1220767,2173346,

%U 3871747,6894873,12276852,21866387,38941846,69344928,123500513,219943018,391676701,697538335

%N p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = 1 - S - S^3.

%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 A291728 for a guide to related sequences.

%H Clark Kimberling, <a href="/A292320/b292320.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 4, -2, 0, -3, 1, 0, 1)

%F G.f.: -((1 + x^2 - 2 x^3 + x^6)/(-1 + x + 4 x^3 - 2 x^4 - 3 x^6 + x^7 + x^9)).

%F a(n) = a(n-1) + 4*a(n-3) - 2*a(n-4) - 3*a(n-6) + a(n-7) + a(n-9) for n >= 10.

%t z = 60; s = x/(x - x^3); p = 1 - s - s^3;

%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A079978  *)

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292320 *)

%Y Cf. A079978, A290616.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Sep 14 2017