OFFSET
0,4
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 A292479 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 (9, -36, 85, -123, 129, -83, 36, -9, 1)
FORMULA
G.f.: -((x^2 (1 + x)^3)/((-1 + 4 x - 2 x^2 + x^3) (1 - 5 x + 14 x^2 - 18 x^3 + 18 x^4 - 7 x^5 + x^6))).
a(n) = 9*a(n-1) - 36*a(n-2) + 85*a(n-3) - 123*a(n-4) + 129*a(n-5) - 83*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n >= 10.
MATHEMATICA
z = 60; s = x (x + 1)/(1 - x)^3; p = 1 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292532 *)
LinearRecurrence[{9, -36, 85, -123, 129, -83, 36, -9, 1}, {0, 0, 1, 12, 75, 329, 1158, 3606, 10971}, 30] (* Harvey P. Dale, Sep 27 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 04 2017
STATUS
approved