A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3.

1, 3, 9, 26, 74, 210, 596, 1692, 4804, 13640, 38728, 109960, 312208, 886448, 2516880, 7146144, 20289952, 57608992, 163568448, 464417728, 1318615104, 3743926400, 10630080640, 30181847168, 85694918912, 243312448256, 690833811712, 1961475291648, 5569190816256

Offset: 0

Author

Clark Kimberling, Aug 22 2017

Keywords

• nonn
• easy

Comments

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).

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:

p(S) t(1,1,1,1,1,...)

1 - S A000079

1 - S^2 A000079

1 - S^3 A024495

1 - S^4 A000749

1 - S^5 A139761

1 - S^6 A290993

1 - S^7 A290994

1 - S^8 A290995

1 - S - S^2 A001906

1 - S - S^3 A116703

1 - S - S^4 A290996

1 - S^3 - S^6 A290997

1 - S^2 - S^3 A095263

1 - S^3 - S^4 A290998

1 - 2 S^2 A052542

1 - 3 S^2 A002605

1 - 4 S^2 A015518

1 - 5 S^2 A163305

1 - 6 S^2 A290999

1 - 7 S^2 A291008

1 - 8 S^2 A291001

(1 - S)^2 A045623

(1 - S)^3 A058396

(1 - S)^4 A062109

(1 - S)^5 A169792

(1 - S)^6 A169793

(1 - S^2)^2 A024007

1 - 2 S - 2 S^2 A052530

1 - 3 S - 2 S^2 A060801

(1 - S)(1 - 2 S) A053581

(1 - 2 S)(1 - 3 S) A291002

(1 - S)(1 - 2 S)(1 - 3 S)(1 - 4 S) A291003

(1 - 2 S)^2 A120926

(1 - 3 S)^2 A291004

1 + S - S^2 A000045 (Fibonacci numbers starting with -1)

1 - S - S^2 - S^3 A291000

1 - S - S^2 - S^3 - S^4 A291006

1 - S - S^2 - S^3 - S^4 - S^5 A291007

1 - S^2 - S^4 A290990

(1 - S)(1 - 3 S) A291009

(1 - S)(1 - 2 S)(1 - 3 S) A291010

(1 - S)^2 (1 - 2 S) A291011

(1 - S^2)(1 - 2 S) A291012

(1 - S^2)^3 A291013

(1 - S^3)^2 A291014

1 - S - S^2 + S^3 A045891

1 - 2 S - S^2 + S^3 A291015

1 - 3 S + S^2 A136775

1 - 4 S + S^2 A291016

1 - 5 S + S^2 A291017

1 - 6 S + S^2 A291018

1 - S - S^2 - S^3 + S^4 A291019

1 - S - S^2 - S^3 - S^4 + S^5 A291020

1 - S - S^2 - S^3 + S^4 + S^5 A291021

1 - S - 2 S^2 + 2 S^3 A175658

1 - 3 S^2 + 2 S^3 A291023

(1 - 2 S^2)^2 A291024

(1 - S^3)^3 A291143

(1 - S - S^2)^2 A209917

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4, -4, 2)

Crossrefs

Cf. A000012, A289780.

Programs

Mathematica

z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291000 *)

Formula

G.f.: (-1 + x - x^2)/(-1 + 4 x - 4 x^2 + 2 x^3).

a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3) for n >= 4.