A291404 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - S^2 - 2 S^4.

0, 1, 2, 4, 12, 23, 42, 89, 188, 404, 856, 1763, 3652, 7641, 16030, 33612, 70252, 146623, 306334, 640637, 1340024, 2802056, 5857264, 12243403, 25596040, 53515853, 111889138, 233922392, 489039852, 1022399607, 2137493106, 4468804953, 9342779572, 19532522316

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 (0, 1, 2, 3, 8, 12, 8, 2)

Crossrefs

Cf. A019590, A291382.

Programs

Mathematica

z = 60; s = x + x^2; p = 1 - s^2 - 2 s^4;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291404 *)

Formula

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

a(n) = a(n-2) + 2*a(n-3) + 3*a(n-4) + 8*a(n-5) + 12*a(n-6) + 8*a(n-7) + 2*a(n-8) for n >= 9.