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