OFFSET
0,3
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 (1, 4, -3, -5, 3, 4, -1, -1)
FORMULA
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 5*a(n-4) + 3*a(n-5) + 4*a(n-6) - a(n-7) - a(n-8) for n >= 9.
G.f.: (1 + x - x^2)*(1 - x - x^2 + x^3 + x^4) / ((1 - x - 2*x^2 + x^4)*(1 - 2*x^2 + x^3 + x^4)). - Colin Barker, Aug 25 2017
MATHEMATICA
PROG
(PARI) Vec((1 + x - x^2)*(1 - x - x^2 + x^3 + x^4) / ((1 - x - 2*x^2 + x^4)*(1 - 2*x^2 + x^3 + x^4)) + O(x^30)) \\ Colin Barker, Aug 25 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 24 2017
STATUS
approved