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A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3. 57
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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)

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.

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 *)

CROSSREFS

Cf. A000012, A289780.

Sequence in context: A127911 A116423 A077845 * A276068 A171277 A289806

Adjacent sequences:  A290997 A290998 A290999 * A291001 A291002 A291003

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 22 2017

STATUS

approved

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Last modified December 12 16:06 EST 2018. Contains 318077 sequences. (Running on oeis4.)