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 A291000 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,-9,7,-1).
FORMULA
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - a(n-4) for n >= 4.
G.f.: (1 - 3*x + 3*x^2) / (1 - 5*x + 9*x^2 - 7*x^3 + x^4). - Colin Barker, Aug 22 2017
MATHEMATICA
z = 60; s = x/(1-x); p = 1 -s -s^4;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290996 *)
LinearRecurrence[{5, -9, 7, -1}, {1, 2, 4, 9}, 60] (* G. C. Greubel, Apr 13 2023 *)
PROG
(PARI) Vec((1 - 3*x + 3*x^2) / (1 - 5*x + 9*x^2 - 7*x^3 + x^4) + O(x^50)) \\ Colin Barker, Aug 22 2017
(Magma) I:=[1, 2, 4, 9]; [n le 4 select I[n] else 5*Self(n-1) -9*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..51]]; // G. C. Greubel, Apr 13 2023
(SageMath)
@CachedFunction
def a(n): # a = A290996
if(n<4): return (1, 2, 4, 9)[n]
else: return 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - a(n-4)
[a(n) for n in range(61)] # G. C. Greubel, Apr 13 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 22 2017
STATUS
approved