OFFSET
0,7
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 (0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1)
FORMULA
G.f.: -((x (1 + x)^2 (1 - x + x^2 - x^3 + x^4)^2)/((-1 + x + x^6) (1 + x + x^6))).
a(n) = a(n-2) + 2*a(n-7) + a(n-12) for n >= 13.
MATHEMATICA
z = 60; s = x + x^4; p = 1 - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292403 *)
LinearRecurrence[{0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1}, {0, 1, 0, 1, 0, 1, 2, 1, 4, 1, 6, 2}, 60] (* Vincenzo Librandi, Oct 01 2017 *)
PROG
(Magma) I:=[0, 1, 0, 1, 0, 1, 2, 1, 4, 1, 6, 2]; [n le 12 select I[n] else Self(n-2)+2*Self(n-7)+Self(n-12): n in [1..60]]; // Vincenzo Librandi, Oct 01 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 30 2017
STATUS
approved