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..999
Index entries for linear recurrences with constant coefficients, signature (14,-71,154,-120).
FORMULA
G.f.: (10 - 65*x + 150*x^2 - 119*x^3)/(1 - 14*x + 71*x^2 - 154*x^3 + 120*x^4).
a(n) = 14*a(n-1) - 71*a(n-2) + 154*a(n-3) - 120*a(n-4) for n >= 5.
a(n) = (-2^n + 16*3^(1+n) - 243*4^n + 256*5^n) / 6. - Colin Barker, Aug 23 2017
MATHEMATICA
z = 60; s = x/(1-x); p = (1 - s)(1 - 2 s)(1 - 3 s)(1 - 4 s);
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291003 *)
LinearRecurrence[{14, -71, 154, -120}, {10, 75, 490, 2956}, 41] (* G. C. Greubel, May 23 2023 *)
PROG
(Magma) [(-2^n +48*3^n -243*4^n +256*5^n)/6: n in [0..40]]; // G. C. Greubel, May 23 2023
(SageMath) [(-2^n +48*3^n -243*4^n +256*5^n)//6 for n in range(41)] # G. C. Greubel, May 23 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 22 2017
STATUS
approved