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 A290890 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 (12, -63, 196, -414, 636, -731, 636, -414, 196, -63, 12, -1)
FORMULA
a(n) = 12*a(n-1) - 63*a(n-2) + 196*a(n-3) - 414*a(n-4) + 636*a(n-5) - 731*a(n-6) + 636*a(n-7) - 414*a(n-8) + 196*a(n-9) - 63*a(n-10) + 12*a(n-11) - a(n-12).
G.f.: x*(3 - 24*x + 81*x^2 - 156*x^3 + 193*x^4 - 156*x^5 + 81*x^6 - 24*x^7 + 3*x^8) / ((1 - 3*x + x^2)^3*(1 - x + x^2)^3). - Colin Barker, Aug 19 2017
MATHEMATICA
PROG
(PARI) concat(0, Vec(x*(3 - 24*x + 81*x^2 - 156*x^3 + 193*x^4 - 156*x^5 + 81*x^6 - 24*x^7 + 3*x^8) / ((1 - 3*x + x^2)^3*(1 - x + x^2)^3) + O(x^30))) \\ Colin Barker, Aug 19 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 19 2017
STATUS
approved