OFFSET
0,4
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,-15,21,-18,9,-1).
FORMULA
a(n) = 6*a(n-1) - 15*a(n-2) + 21*a(n-3) - 18*a(n-4) + 9*a(n-5) - a(n-6) for n >= 7.
G.f.: x^2*(1 - 3*x + 3*x^2) / (1 - 6*x + 15*x^2 - 21*x^3 + 18*x^4 - 9*x^5 + x^6). - Colin Barker, Aug 22 2017
MATHEMATICA
z = 60; s = x/(1-x); p= 1 -s^3 -s^6;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290997 *)
LinearRecurrence[{6, -15, 21, -18, 9, -1}, {0, 0, 1, 3, 6, 12}, 40] (* G. C. Greubel, Apr 14 2023 *)
PROG
(PARI) concat(vector(2), Vec(x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) + O(x^50))) \\ Colin Barker, Aug 22 2017
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0, 0] cat Coefficients(R!( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) )); // G. C. Greubel, Apr 14 2023
(SageMath)
def A290997_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) ).list()
A290997_list(40) # G. C. Greubel, Apr 14 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 22 2017
STATUS
approved