A289845 p-INVERT of A079977, where p(S) = 1 - S - S^2.

1, 2, 4, 9, 19, 43, 91, 202, 433, 952, 2055, 4494, 9737, 21236, 46099, 100403, 218164, 474833, 1032256, 2245929, 4883690, 10623848, 23103985, 50255443, 109298635, 237734446, 517055409, 1124617945, 2446001258, 5320100761, 11571106298, 25167245524, 54738437517

Offset

0,2

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 A289780 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..999

Index entries for linear recurrences with constant coefficients, signature (1, 3, -1, 1, -1, -2, 0, -1)

Formula

G.f.: (1 + x - x^2 - x^4)/(1 - x - 3 x^2 + x^3 - x^4 + x^5 + 2 x^6 + x^8).

a(n) = a(n-1) + 3*a(n-2) - a(n-3) + a(n-4) - a(n-5) - 2*a(n-6) - a(n-8).

Mathematica

z = 60; s = -x/(x^4 + x^2 - 1); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A079977 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (*A289845*)
LinearRecurrence[{1, 3, -1, 1, -1, -2, 0, -1}, {1, 2, 4, 9, 19, 43, 91, 202}, 40] (* Harvey P. Dale, Jan 16 2019 *)

Crossrefs

Cf. A079977, A289780.

Sequence in context: A193021 A112569 A301706 * A101463 A319379 A347011

Adjacent sequences: A289842 A289843 A289844 * A289846 A289847 A289848

Keyword

nonn,easy

Author

Clark Kimberling, Aug 14 2017

Status

approved