A292321 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = 1 - S^2 - S^3.

0, 1, 1, 1, 4, 5, 7, 17, 23, 38, 75, 109, 190, 339, 524, 917, 1563, 2519, 4360, 7305, 12056, 20621, 34407, 57452, 97423, 162672, 272961, 460454, 770281, 1294575, 2177777, 3649129, 6134192, 10306017, 17287962, 29054244, 48790024, 81894794, 137592982

Offset: 0

Author

Clark Kimberling, Sep 15 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 (0, 1, 4, 0, -1, -3, 0, 0, 1)

Crossrefs

Cf. A079978, A292320.

Programs

Mathematica

z = 60; s = x/(x - x^3); p = 1 - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A079978  *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292321 *)

Formula

G.f.: (x (-1 - x + x^3))/(-1 + x^2 + 4 x^3 - x^5 - 3 x^6 + x^9).

a(n) = a(n-2) + 4*a(n-3) - a(n-5) - 3*a(n-6) + a(n-9) for n >= 10.