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).
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -8, 11, -8, 4, -1)
FORMULA
G.f.: (1 - 2 x + 4 x^2 - 2 x^3 + x^4)/(1 - 4 x + 8 x^2 - 11 x^3 + 8 x^4 - 4 x^5 + x^6).
a(n) = 4*a(n-1) - 8*a(n-2) + 11*a(n-3) - 8*a(n-4) + 4*a(n-5) - a(n-6) for n >= 7.
MATHEMATICA
z = 60; s = x/(1 - x + x^2); p = 1 - s - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A010892 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292478 *)
LinearRecurrence[{4, -8, 11, -8, 4, -1}, {1, 2, 4, 9, 19, 36}, 40] (* Vincenzo Librandi, Oct 01 2017 *)
PROG
(Magma) I:=[1, 2, 4, 9, 19, 36]; [n le 6 select I[n] else 4*Self(n-1)-8*Self(n-2)+11*Self(n-3)-8*Self(n-4)+4*Self(n-5)-Self(n-6): n in [1..35]]; // Vincenzo Librandi, Oct 01 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 30 2017
STATUS
approved