OFFSET
0,5
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 A291219 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 (0,3, 1,-3,0,1)
FORMULA
G.f.: -(x^2/((-1 + x + x^2) (1 + x - x^2 - x^3 + x^4))).
a(n) = 3*a(n-2) + a(n-3) - 3*a(n-4) + a(n-6) for n >= 7.
MATHEMATICA
z = 60; s = x/(1 - x^2); p = 1 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291217 *)
LinearRecurrence[{0, 3, 1, -3, 0, 1}, {0, 0, 1, 0, 3, 1}, 50] (* Vincenzo Librandi, Aug 25 2017 *)
PROG
(Magma) I:=[0, 0, 1, 0, 3, 1]; [n le 6 select I[n] else 3*Self(n-2)+Self(n-3)-3*Self(n-4)+Self(n-6): n in [1..45]]; // Vincenzo Librandi, Aug 25 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 24 2017
STATUS
approved