A291009 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S)*(1 - 3*S).

4, 17, 70, 284, 1144, 4592, 18400, 73664, 294784, 1179392, 4718080, 18873344, 75495424, 301985792, 1207951360, 4831821824, 19327320064, 77309345792, 309237514240, 1236950319104, 4947801800704, 19791208251392, 79164835102720, 316659344605184, 1266637386809344

Offset: 0

Author

Clark Kimberling, Aug 23 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 A291000 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 (6,-8).

Crossrefs

Cf. A000012, A033453, A289780, A291000.

Programs

Magma

[2^(n-1)*(9*2^n -1): n in [0..40]]; // G. C. Greubel, Jun 04 2023

Mathematica

z = 60; s = x/(1-x); p = (1-s)(1-3s);
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291009 *)
LinearRecurrence[{6,-8}, {4,17}, 40] (* G. C. Greubel, Jun 04 2023 *)

PARI

Vec((4-7*x)/((1-2*x)*(1-4*x)) + O(x^30)) \\ Colin Barker, Aug 23 2017

SageMath

A291009=BinaryRecurrenceSequence(6,-8,4,17)
[A291009(n) for n in range(41)] # G. C. Greubel, Jun 04 2023

Formula

a(n) = 6*a(n-1) - 8*a(n-2) for n >= 3.

From Colin Barker, Aug 23 2017: (Start)

G.f.: (4 - 7*x) / ((1 - 2*x)*(1 - 4*x)).

a(n) = 2^(n-1) * (9*2^n - 1).

(End)