A024007 -id:A024007 - cp's OEIS frontend

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

Clark Kimberling, Aug 22 2017

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

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

Cf. A000012, A289780.

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 *)

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.

A258837 a(n) = 1 - n^2.

Original entry on oeis.org

1, 0, -3, -8, -15, -24, -35, -48, -63, -80, -99, -120, -143, -168, -195, -224, -255, -288, -323, -360, -399, -440, -483, -528, -575, -624, -675, -728, -783, -840, -899, -960, -1023, -1088, -1155, -1224, -1295, -1368, -1443, -1520, -1599, -1680, -1763, -1848

Offset: 0

Vincenzo Librandi, Jun 12 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A067998, A131386, A007531.

Sequences of the type 1-n^k: A024000 (k=1), this sequence (k=2), A024001 (k=3), A024002 (k=4), A024003 (k=5), A024004 (k=6), A024005 (k=7), A024006 (k=8), A024007 (k=9), A024008 (k=10), A024009 (k=11), A024010 (k=12).

Magma
```
[1-n^2: n in [0..50]];
```

I:=[1,0,-3]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Table[1 - n^2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 0, -3}, 50]

my(x='x+O('x^50)); Vec((1-3*x)/(1-x)^3) \\ G. C. Greubel, May 11 2017

G.f.: (1-3*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = -A067998(n+1). - Joerg Arndt, Jun 13 2015
a(n) = (-1)^n*A131386(n+1). - Bruno Berselli, Jun 15 2015
E.g.f.: (1 - x - x^2)*exp(x). - G. C. Greubel, May 11 2017
Sum_{n>=2} 1/a(n) = -3/4. - Amiram Eldar, Feb 17 2023

A258810 a(n) = n^9 - 1.

Original entry on oeis.org

0, 511, 19682, 262143, 1953124, 10077695, 40353606, 134217727, 387420488, 999999999, 2357947690, 5159780351, 10604499372, 20661046783, 38443359374, 68719476735, 118587876496, 198359290367, 322687697778, 511999999999, 794280046580, 1207269217791, 1801152661462

Offset: 1

Vincenzo Librandi, Jun 11 2015

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A024007, similar sequences listed in A258807.

Magma
```
[n^9-1: n in [1..40]];
```

I:=[0,511,19682, 262143,1953124,10077695,40353606,134217727,387420488, 999999999]; [n le 10 select I[n] else 10*Self(n-1)-45*Self(n-2)+120*Self(n-3)-210*Self(n-4)+252*Self(n-5)-210*Self(n-6)+120*Self(n-7)-45*Self(n-8)+10*Self(n-9)-Self(n-10): n in [1..40]];

Table[n^9 - 1, {n, 33}] (* or *) LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 511, 19682, 262143, 1953124, 10077695, 40353606, 134217727, 387420488, 999999999}, 40]

G.f.: x^2*(511 + 14572*x + 88318*x^2 + 156064*x^3 + 88360*x^4 + 14524*x^5 + 538*x^6 - 8*x^7 + x^8)/(1-x)^10.

a(n) = 10*a(n-1)-45*a(n-2)+120*a(n-3)-210*a(n-4)+252*a(n-5)-210*a(n-6)+120*a(n-7)-45*a(n-8)+10*a(n-9)-a(n-10).

cp's OEIS Frontend

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

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A258837 a(n) = 1 - n^2.

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A258810 a(n) = n^9 - 1.

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