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 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 (5, -1)
FORMULA
EXAMPLE
s = (1,2,3,4,...), p(S) = 1-3*S;
S(x) = x + 2 x^2 + 3 x^3 + ... ;
p(S(x)) = 1 - 3(x + 2 x^2 + 3 x^3 + ...);
1/p(S(x)) = 1 + 3 x + 15 x^2 + 72 x^3 + ... ;
(-p(0) + 1/p(S(x)))/x = 3 + 15 x + 72 x^2 + ... ;
t(s) = (3, 15, 72, ...), with offset 0.
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 17 2017
STATUS
approved