OFFSET
0,1
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 (6, -7, -6, -1)
FORMULA
G.f.: -((3 (-2 + 3 x + 2 x^2))/(-1 + 3 x + x^2)^2).
a(n) = 6*a(n-1) - 7*a(n-2) -6*a(n-3) - a(n-4) for n >= 5.
a(n) = 3 * (((3-sqrt(13))/2)^n*(-3+sqrt(13))*(-39+17*sqrt(13)-39*n) + 2^(-n)*(3+sqrt(13))^(1+n)*(39+17*sqrt(13)+39*n)) / 338. - Colin Barker, Aug 26 2017
MATHEMATICA
z = 60; s = x/(1 - x^2); p = (1 - 3 s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291232 *)
u/3 (* A291265 *)
LinearRecurrence[{6, -7, -6, -1}, {6, 27, 114, 459}, 25] (* Vincenzo Librandi, Aug 28 2017 *)
PROG
(PARI) Vec(3*(2 + x)*(1 - 2*x) / (1 - 3*x - x^2)^2 + O(x^30)) \\ Colin Barker, Aug 26 2017
(Magma) I:=[6, 27, 114, 459]; [n le 4 select I[n] else 6*Self(n-1)-7*Self(n-2)-6*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Aug 28 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 26 2017
STATUS
approved