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 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,-6,1).
FORMULA
G.f.: (2 - 3*x)/(1 - 5*x + 6*x^2 - x^3).
a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3) n >= 4.
MATHEMATICA
z = 60; s = x/(1-x); p = (1 - s^3)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291015 *)
LinearRecurrence[{5, -6, 1}, {2, 7, 23}, 50] (* G. C. Greubel, Jun 06 2023 *)
PROG
(Magma) I:=[2, 7, 23]; [n le 3 select I[n] else 5*Self(n-1) -6*Self(n-2) +Self(n-3): n in [1..50]]; // G. C. Greubel, Jun 06 2023
(SageMath)
@CachedFunction
def a(n): # a = A291015
if (n<3): return (2, 7, 23)[n]
else: return 5*a(n-1) - 6*a(n-2) + a(n-3)
[a(n) for n in range(51)] # G. C. Greubel, Jun 06 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 23 2017
STATUS
approved