OFFSET
0,9
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 (8,-28,56,-70,56,-28,8).
FORMULA
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) for n >= 9.
G.f.: x^7 / ((1 - 2*x)*(1 - 2*x + 2*x^2)*(1 - 4*x + 6*x^2 - 4*x^3 + 2*x^4)). - Colin Barker, Aug 22 2017
G.f.: x^7/((1-x)^8 - x^8). - G. C. Greubel, Apr 11 2023
MATHEMATICA
PROG
(PARI) concat(vector(7), Vec(x^7 / ((1 - 2*x)*(1 - 2*x + 2*x^2)*(1 - 4*x + 6*x^2 - 4*x^3 + 2*x^4)) + O(x^50))) \\ Colin Barker, Aug 22 2017
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!( x^7/((1-x)^8 - x^8) )); // G. C. Greubel, Apr 11 2023
(SageMath)
def A290995_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^7/((1-x)^8 - x^8) ).list()
A290995_list(60) # G. C. Greubel, Apr 11 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 22 2017
STATUS
approved