A291417 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 4 S + 2 S^2.

4, 18, 76, 322, 1360, 5744, 24256, 102428, 432528, 1826456, 7712656, 32568568, 137528704, 580748416, 2452351488, 10355650832, 43729255232, 184657419808, 779760883392, 3292730050592, 13904353779456, 58714516845824, 247936332973056, 1046971490364864

Offset: 0

Author

Clark Kimberling, Sep 07 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 A291382 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 (4, 2, -4, -2)

Crossrefs

Cf. A019590, A291382, A291462.

Programs

Mathematica

z = 60; s = x + x^2; p = 1 - 4 s + 2 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291417 *)
u / 2  (*A291462)

Formula

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

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