cp's OEIS Frontend

A291397 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - S - S^3.

%I A291397 #4 Sep 06 2017 21:15:59
%S A291397 1,2,4,10,22,46,97,209,452,972,2084,4472,9608,20645,44345,95238,
%T A291397 204552,439366,943734,2027046,4353861,9351633,20086392,43143592,
%U A291397 92668072,199041584,427521184,918272425,1972356577,4236422746,9099408124,19544609858,41979848918
%N A291397 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - S - S^3.
%C A291397 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 A291397 See A291382 for a guide to related sequences.
%H A291397 Clark Kimberling, <a href="/A291397/b291397.txt">Table of n, a(n) for n = 0..1000</a>
%H A291397 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 1, 3, 3, 1)
%F A291397 G.f.: -(((1 + x) (1 + x^2 + 2 x^3 + x^4))/((1 + x^2 + x^3) (-1 + x + 2 x^2 + x^3))).
%F A291397 a(n) = a(n-1) + a(n-2) + a(n-3) + 3*a(n-4) + 3*a(n-5) + a(n-6) for n >= 7.
%t A291397 z = 60; s = x + x^2; p = 1 - s - s^3;
%t A291397 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
%t A291397 u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291397 *)
%Y A291397 Cf. A019590, A291382.
%K A291397 nonn,easy
%O A291397 0,2
%A A291397 _Clark Kimberling_, Sep 06 2017