A291728 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2.

1, 2, 4, 9, 17, 35, 70, 142, 285, 576, 1160, 2340, 4716, 9510, 19171, 38653, 77926, 157110, 316747, 638599, 1287479, 2595698, 5233196, 10550681, 21271280, 42885152, 86460984, 174314476, 351436368, 708532813, 1428476905, 2879960190, 5806303628, 11706120825

Offset: 0

Author

Clark Kimberling, Sep 08 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 = (1,0,1,0,0,0,0,...) = A154272, in some cases t(1,0,1,0,0,0,0,...) is a shifted (or differently indexed) version of the indicated sequence:

***

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

1 - S A000930 (Narayana's cows sequence)

1 - S^2 A002478 (except for 0's)

1 - S^3 A291723

1 - S^5 A291724

(1 - S)^2 A291725

(1 - S)^3 A291726

(1 - S)^4 A291727

1 - S - S^2 A291728

1 - 2S - S^2 A291729

1 - 2S - 2S^2 A291730

(1 - 2S)^2 A291732

(1 - S)(1 - 2S) A291734

1 - S - S^3 A291735

1 - S^2 - S^3 A291736

1 - S - S^2 - S^3 A291737

1 - S - S^4 A291738

1 - S^3 - S^6 A291739

(1 - S)(1 - S^2) A291740

(1 - S)(1 + S^2) A291741

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,1,2,0,1).

Crossrefs

Cf. A154272, A290890, A291000, A291382, A291219, A291382.

Programs

Mathematica

z = 60; s = x + x^3; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291728 *)

Formula

G.f.: (-1 - x - x^2 - 2 x^3 - x^5)/(-1 + x + x^2 + x^3 + 2 x^4 + x^6).

a(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) + a(n-6) for n >= 7.