A291035 -id:A291035 - cp's OEIS frontend

A290616 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = 1 - 2 S - S^2.

2, 5, 12, 31, 80, 205, 526, 1350, 3464, 8889, 22810, 58532, 150198, 385420, 989018, 2537899, 6512450, 16711463, 42882940, 110041025, 282373998, 724594076, 1859365870, 4771280299, 12243483684, 31417750230, 80620439004, 206878440932, 530866488090

Offset: 0

Clark Kimberling, Sep 14 2017

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.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, 2, -2, 0, -1)

Cf. A079978, A289918, A291035.

z = 60; s = x/(x - x^3); p = 1 - 2 s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A079978  *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290616 *)

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

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

cp's OEIS Frontend

A290616 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = 1 - 2 S - S^2.

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