A291233 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S - S^2 - S^3.

1, 2, 5, 11, 26, 58, 134, 303, 693, 1576, 3593, 8184, 18645, 42480, 96773, 220481, 502290, 1144350, 2607062, 5939501, 13531493, 30827806, 70232669, 160005808, 364529269, 830479602, 1892019493, 4310445875, 9820165646, 22372546322, 50969693930, 116120429167

Offset: 0

Author

Clark Kimberling, Aug 26 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 A291219 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, 4, -1, -4, 1, 1)

Crossrefs

Cf. A000035, A291219.

Programs

Mathematica

z = 60; s = x/(1 - x^2); p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000035 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291233 *)

Formula

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

a(n) = a(n-1) + 4*a(n-2) - a(n-3) - 4*a(n-4) + a(n-5) + a(n-6) for n >= 7.