A291007 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3 - S^4 - S^5.

1, 3, 9, 27, 81, 242, 720, 2137, 6337, 18789, 55715, 165232, 490058, 1453493, 4311025, 12786359, 37923789, 112480082, 333610072, 989469949, 2934716101, 8704215281, 25816251319, 76569665176, 227101665034, 673571786617, 1997779058053, 5925309279179

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,-13,14,-7,2).

Crossrefs

Cf. A000012, A289780, A291000.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1 -3*x +4*x^2 -2*x^3 +x^4)/(1 -6*x +13*x^2 -14*x^3 +7*x^4 -2*x^5) )); // G. C. Greubel, Jun 01 2023

Mathematica

z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3 - s^4 - s^5;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291007 *)
LinearRecurrence[{6,-13,14,-7,2},{1,3,9,27,81},30] (* Harvey P. Dale, Apr 07 2019 *)

PARI

Vec((1 -3*x +4*x^2 -2*x^3 +x^4)/(1 -6*x +13*x^2 -14*x^3 +7*x^4 - 2*x^5) + O(x^30)) \\ Colin Barker, Aug 23 2017

SageMath

def A291007_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1 -3*x +4*x^2 -2*x^3 +x^4)/(1 -6*x +13*x^2 -14*x^3 +7*x^4 -2*x^5) ).list()
A291007_list(40) # G. C. Greubel, Jun 01 2023

Formula

a(n) = 6*a(n-1) - 13*a(n-2) + 14*a(n-3) - 7*a(n-4) + 2*a(n-5) for n >= 6.

G.f.: (1 - 3*x + 4*x^2 - 2*x^3 + x^4) / (1 - 6*x + 13*x^2 - 14*x^3 + 7*x^4 - 2*x^5). - Colin Barker, Aug 23 2017