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).
In the following guide to p-INVERT sequences using s = (1,0,1,0,0,0,0,...) = A154272, in some cases t(1,0,1,0,0,0,0,...) is a shifted (or differently indexed) version of the indicated sequence:
***
p(S) t(1,0,1,0,0,0,0,...)
1 - S A000930 (Narayana's cows sequence)
1 - S^2 A002478 (except for 0's)
1 - S^3 A291723
1 - S^5 A291724
(1 - S)^2 A291725
(1 - S)^3 A291726
(1 - S)^4 A291727
1 - S - S^2 A291728
1 - 2S - S^2 A291729
1 - 2S - 2S^2 A291730
(1 - 2S)^2 A291732
(1 - S)(1 - 2S) A291734
1 - S - S^3 A291735
1 - S^2 - S^3 A291736
1 - S - S^2 - S^3 A291737
1 - S - S^4 A291738
1 - S^3 - S^6 A291739
(1 - S)(1 - S^2) A291740
(1 - S)(1 + S^2) A291741
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 2, 0, 1)
FORMULA
G.f.: (-1 - x - x^2 - 2 x^3 - x^5)/(-1 + x + x^2 + x^3 + 2 x^4 + x^6).
a(n) = a(n-1) + a(n-2) + a(n-3) + 2*(a(n-4) + a(n-6) for n >= 7.
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 08 2017
STATUS
approved