A289919 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = (1 - S)^2.

2, 3, 4, 7, 12, 19, 30, 48, 76, 119, 186, 290, 450, 696, 1074, 1653, 2538, 3889, 5948, 9081, 13842, 21068, 32022, 48609, 73700, 111618, 168868, 255232, 385410, 581479, 876576, 1320411, 1987516, 2989583, 4493910, 6750968, 10135584, 15208443, 22807902

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 (2, -1, 2, -2, 0, -1)

Crossrefs

Cf. A079978, A289918.

Programs

Mathematica

z = 60; s = x/(x - x^3); p = (1 - s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A079978  *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A289919 *)

Formula

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

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