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 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).
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
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 *)
PROG
(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
(Magma) R<x>:=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
(SageMath)
def A291007_list(prec):
P.<x> = 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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 23 2017
STATUS
approved