A291729 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2.

2, 5, 14, 39, 106, 290, 794, 2173, 5946, 16272, 44530, 121860, 333480, 912597, 2497400, 6834349, 18702782, 51181767, 140063294, 383295214, 1048920220, 2870460125, 7855260268, 21496593296, 58827270844, 160985870984, 440551640160, 1205607339709, 3299247863502

Offset: 0

Author

Clark Kimberling, Sep 11 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).

See A291728 for a guide to related sequences.

Links

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

Crossrefs

Cf. A154272, A291728.

Programs

Mathematica

z = 60; s = x + x^3; p = 1 - 2 s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291729 *)

Formula

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

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