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).
See A291000 for a guide to related sequences.
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (4, -2, -4, -1)
FORMULA
G.f.: -((4 (-x + 2 x^2))/(-1 + 2 x + x^2)^2).
a(n) = 4*a(n-1) - 2 a(n-2) - 4*a(n-3) - a(n-4) for n >= 5.
a(n) = 4*A291142(n) for n >= 0.
a(n) = ((1+sqrt(2))^n*(3*sqrt(2) + 2*(-1+sqrt(2))*n) - (1-sqrt(2))^n*(3*sqrt(2) + 2*(1+sqrt(2))*n)) / 4. - Colin Barker, Aug 24 2017
MATHEMATICA
PROG
(PARI) concat(0, Vec(4*x*(1 - 2*x) / (1 - 2*x - x^2)^2 + O(x^30))) \\ Colin Barker, Aug 24 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 24 2017
STATUS
approved