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A291382 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2. 41
2, 7, 22, 70, 222, 705, 2238, 7105, 22556, 71608, 227332, 721705, 2291178, 7273743, 23091762, 73308814, 232731578, 738846865, 2345597854, 7446508273, 23640235416, 75050038224, 238259397096, 756395887969, 2401310279090, 7623377054503, 24201736119310 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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 = (1,1,0,0,0,...) = A019590, in some cases t(1,1,0,0,0,...) is a shifted version of the cited sequence:

p(S)                     t(1,1,0,0,0,...)

1 - S                       A000045 (Fibonacci numbers)

1 - S^2                     A094686

1 - S^3                     A115055

1 - S^4                     A291379

1 - S^5                     A281380

1 - S^6                     A281381

1 - 2 S                     A002605

1 - 3 S                     A125145

(1 - S)^2                   A001629

(1 - S)^3                   A001628

(1 - S)^4                   A001629

(1 - S)^5                   A001873

(1 - S)^6                   A001874

1 - S - S^2                 A123392

1 - 2 S - S^2               A291382

1 - S - 2 S^2               A124861

1 - 2 S - S^2               A291383

(1 - 2 S)^2                 A073388

(1 - 3 S)^2                 A291387

(1 - 5 S)^2                 A291389

(1 - 6 S)^2                 A291391

(1 - S)(1 - 2 S)            A291393

(1 - S)(1 - 3 S)            A291394

(1 - 2 S)(1 - 3 S)          A291395

(1 - S)(1 - 2 S)            A291393

(1 - S)(1 - 2 S)(1 - 3 S)   A291396

1 - S - S^3                 A291397

1 - S^2 - S^3               A291398

1 - S - S^2 - S^3           A186812

1 - S - S^2 - S^3 - S^4     A291399

1 - S^2 - S^4               A291400

1 - S - S^4                 A291401

1 - S^3 - S^4               A291402

1 - 2 S^2 - S^4             A291403

1 - S^2 - 2 S^4             A291404

1 - 2 S^2 - 2 S^4           A291405

1 - S^3 - S^6               A291407

(1 - S)(1 - S^2)            A291408

(1 - S^2)(1 - S)^2          A291409

1 - S - S^2 - 2 S^3         A291410

1 - 2 S - S^2 + S^3         A291411

1 - S - 2 S^2 + S^3         A291412

1 - 3 S + S^2 + S^3         A291413

1 - 2 S + S^3               A291414

1 - 3 S + S^2               A291415

1 - 4 S + S^2               A291416

1 - 4 S + 2 S^2             A291417

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2, 3, 2, 1)

FORMULA

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

a(n) = 2*a(n-1) + 3*a(n-2) + 2*a(n-3) + a(n-4) for n >= 5.

MATHEMATICA

z = 60; s = x + x^2; p = 1 - 2 s - s^2;

Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)

Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291382 *)

CROSSREFS

Cf. A019590, A290890, A291000, A291219, A291728, A292479, A292480.

Sequence in context: A094618 A077833 A106438 * A109999 A092690 A030186

Adjacent sequences:  A291379 A291380 A291381 * A291383 A291384 A291385

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Sep 04 2017

STATUS

approved

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Last modified January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)