A291737 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2 - S^3.

1, 2, 5, 11, 25, 54, 121, 267, 591, 1310, 2899, 6422, 14218, 31486, 69722, 154389, 341881, 757050, 1676405, 3712200, 8220236, 18202762, 40307892, 89257156, 197649588, 437672056, 969173912, 2146123007, 4752340053, 10523504828, 23303078705, 51601960101

Offset: 0

Author

Clark Kimberling, Sep 11 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 (1, 1, 2, 2, 3, 1, 3, 0, 1)

Crossrefs

Cf. A154272, A291728.

Programs

Mathematica

z = 60; s = x + x^3; p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291737 *)

Formula

G.f.: -(((1 + x^2) (1 - x + x^2) (1 + 2 x + 2 x^2 + x^3 + x^4))/(-1 + x + x^2 + 2 x^3 + 2 x^4 + 3 x^5 + x^6 + 3 x^7 + x^9)).

a(n) = a(n-1) + a(n-2) + 2*a(n-3) + 2*a(n-4) + 3*a(n-5) + a(n-6) + 3*a(n-7) + a(n-9) for n >= 10.