cp's OEIS Frontend

A291414 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S + S^3.

%I A291414 #4 Sep 11 2017 12:05:59
%S A291414 2,6,15,37,89,212,500,1173,2741,6388,14860,34524,80138,185904,431075,
%T A291414 999280,2315960,5366755,12435075,28810731,66748062,154635086,
%U A291414 358234115,829886167,1922494024,4453566092,10316878400,23899399028,55363614076,128251081293
%N A291414 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S + S^3.
%C A291414 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 A291414 See A291382 for a guide to related sequences.
%H A291414 Clark Kimberling, <a href="/A291414/b291414.txt">Table of n, a(n) for n = 0..1000</a>
%H A291414 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 2, -1, -3, -3, -1)
%F A291414 G.f.: -(((1 + x) (-2 + x^2 + 2 x^3 + x^4))/((-1 + x + x^2) (-1 + x + 2 x^2 + 2 x^3 + x^4))).
%F A291414 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) - 3*a(n-4) - 3*a(n-5) - a(n-6) for n >= 7.
%t A291414 z = 60; s = x + x^2; p = 1 - 2 s + s^3;
%t A291414 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
%t A291414 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291414 *)
%Y A291414 Cf. A019590, A291382.
%K A291414 nonn,easy
%O A291414 0,1
%A A291414 _Clark Kimberling_, Sep 07 2017