A291382 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2.

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

Offset: 0

Author

Clark Kimberling, Sep 04 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 = (1,1,0,0,0,...) = 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)

Crossrefs

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

Programs

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 *)

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.