A291724 -id:A291724 - cp's OEIS frontend

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

Clark Kimberling, Sep 08 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 = (1,0,1,0,0,0,0,...) = 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

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).

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

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 *)

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula