A291006 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3 - S^4.

1, 3, 9, 27, 80, 235, 688, 2013, 5891, 17244, 50482, 147791, 432672, 1266680, 3708288, 10856241, 31782309, 93044665, 272394011, 797450348, 2334585333, 6834643282, 20008841823, 58577124509, 171488162320, 502042223184, 1469759722591, 4302812676894

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..999
• Index entries for linear recurrences with constant coefficients, signature (5,-8,6,-1).

Crossrefs

Cf. A000012, A289780, A291000.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-2*x+2*x^2)/(1-5*x+8*x^2-6*x^3+x^4) )); // G. C. Greubel, Jun 01 2023

Mathematica

z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3 - s^4;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291006 *)
LinearRecurrence[{5,-8,6,-1}, {1,3,9,27}, 41] (* G. C. Greubel, Jun 01 2023 *)

SageMath

def A291006_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x+2*x^2)/(1-5*x+8*x^2-6*x^3+x^4) ).list()
A291006_list(40) # G. C. Greubel, Jun 01 2023

Formula

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

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