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%I #10 Sep 25 2017 02:24:19
%S 1,3,9,26,74,210,596,1692,4804,13640,38728,109960,312208,886448,
%T 2516880,7146144,20289952,57608992,163568448,464417728,1318615104,
%U 3743926400,10630080640,30181847168,85694918912,243312448256,690833811712,1961475291648,5569190816256
%N p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - 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 In the following guide to p-INVERT sequences using s = (1,1,1,1,1,...) = A000012, in some cases t(1,1,1,1,1,...) is a shifted version of the cited sequence:
%C p(S) t(1,1,1,1,1,...)
%C 1 - S A000079
%C 1 - S^2 A000079
%C 1 - S^3 A024495
%C 1 - S^4 A000749
%C 1 - S^5 A139761
%C 1 - S^6 A290993
%C 1 - S^7 A290994
%C 1 - S^8 A290995
%C 1 - S - S^2 A001906
%C 1 - S - S^3 A116703
%C 1 - S - S^4 A290996
%C 1 - S^3 - S^6 A290997
%C 1 - S^2 - S^3 A095263
%C 1 - S^3 - S^4 A290998
%C 1 - 2 S^2 A052542
%C 1 - 3 S^2 A002605
%C 1 - 4 S^2 A015518
%C 1 - 5 S^2 A163305
%C 1 - 6 S^2 A290999
%C 1 - 7 S^2 A291008
%C 1 - 8 S^2 A291001
%C (1 - S)^2 A045623
%C (1 - S)^3 A058396
%C (1 - S)^4 A062109
%C (1 - S)^5 A169792
%C (1 - S)^6 A169793
%C (1 - S^2)^2 A024007
%C 1 - 2 S - 2 S^2 A052530
%C 1 - 3 S - 2 S^2 A060801
%C (1 - S)(1 - 2 S) A053581
%C (1 - 2 S)(1 - 3 S) A291002
%C (1 - S)(1 - 2 S)(1 - 3 S)(1 - 4 S) A291003
%C (1 - 2 S)^2 A120926
%C (1 - 3 S)^2 A291004
%C 1 + S - S^2 A000045 (Fibonacci numbers starting with -1)
%C 1 - S - S^2 - S^3 A291000
%C 1 - S - S^2 - S^3 - S^4 A291006
%C 1 - S - S^2 - S^3 - S^4 - S^5 A291007
%C 1 - S^2 - S^4 A290990
%C (1 - S)(1 - 3 S) A291009
%C (1 - S)(1 - 2 S)(1 - 3 S) A291010
%C (1 - S)^2 (1 - 2 S) A291011
%C (1 - S^2)(1 - 2 S) A291012
%C (1 - S^2)^3 A291013
%C (1 - S^3)^2 A291014
%C 1 - S - S^2 + S^3 A045891
%C 1 - 2 S - S^2 + S^3 A291015
%C 1 - 3 S + S^2 A136775
%C 1 - 4 S + S^2 A291016
%C 1 - 5 S + S^2 A291017
%C 1 - 6 S + S^2 A291018
%C 1 - S - S^2 - S^3 + S^4 A291019
%C 1 - S - S^2 - S^3 - S^4 + S^5 A291020
%C 1 - S - S^2 - S^3 + S^4 + S^5 A291021
%C 1 - S - 2 S^2 + 2 S^3 A175658
%C 1 - 3 S^2 + 2 S^3 A291023
%C (1 - 2 S^2)^2 A291024
%C (1 - S^3)^3 A291143
%C (1 - S - S^2)^2 A209917
%H Clark Kimberling, <a href="/A291000/b291000.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4, -4, 2)
%F G.f.: (-1 + x - x^2)/(-1 + 4 x - 4 x^2 + 2 x^3).
%F a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3) for n >= 4.
%t z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3;
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291000 *)
%Y Cf. A000012, A289780.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Aug 22 2017
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