OFFSET
0,3
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 (6,-15,22,-21,12,-4).
FORMULA
G.f.: x^2*(2 - 6*x + 6*x^2 - 3*x^3)/( (1-2*x)*(1-x+x^2) )^2.
a(n) = 6*a(n-1) - 15*a(n-2) + 22*a(n-3) - 21*a(n-4) + 12*a(n-5) - 4*a(n-6) for n >= 7.
a(n) = (1/9)*( 2^(n-1)*(n + 8) - 3*(A099254(n) - A099254(n-1)) - A010892(n) - 5*A010892(n-1) ). - G. C. Greubel, Jun 05 2023
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] (* A291014 *)
LinearRecurrence[{6, -15, 22, -21, 12, -4}, {0, 0, 2, 6, 12, 23}, 50] (* G. C. Greubel, Jun 05 2023 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0, 0] cat Coefficients(R!( x^2*(2-6*x+6*x^2-3*x^3)/((1-2*x)*(1-x+x^2))^2 )); // G. C. Greubel, Jun 05 2023
(SageMath)
def A291014_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^2*(2-6*x+6*x^2-3*x^3)/((1-2*x)*(1-x+x^2))^2 ).list()
A291014_list(50) # G. C. Greubel, Jun 05 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 23 2017
STATUS
approved