OFFSET
0,1
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).
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)
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 *)
PROG
(PARI) Vec((4-7*x)/((1-2*x)*(1-4*x)) + O(x^30)) \\ Colin Barker, Aug 23 2017
(Magma) [2^(n-1)*(9*2^n -1): n in [0..40]]; // G. C. Greubel, Jun 04 2023
(SageMath)
A291009=BinaryRecurrenceSequence(6, -8, 4, 17)
[A291009(n) for n in range(41)] # G. C. Greubel, Jun 04 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 23 2017
STATUS
approved