A291035 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = 1 - S - 2 S^2.

1, 3, 5, 12, 27, 58, 130, 285, 629, 1389, 3060, 6753, 14892, 32844, 72445, 159775, 352401, 777244, 1714259, 3780930, 8339090, 18392449, 40565829, 89470733, 197333940, 435233685, 959938112, 2117210180, 4669654005, 10299246171, 22715702489, 50101058976

Offset: 0

Author

Clark Kimberling, Sep 14 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 (1,2,2,-1,0,-1).

Crossrefs

Cf. A079978, A289918, A290616.

Programs

Mathematica

z = 60; s = x/(x - x^3); p = 1 - s - 2 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A079978 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291035 *)

Formula

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

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