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..186
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
FORMULA
G.f.: x*(3 - 12*x + 15*x^2 - 6*x^3 + x^4)/(1 - 2*x)^3.
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n >= 7.
a(n) = 2^(n-6) * (60 + 17*n + n^2) for n>2. - Colin Barker, Aug 23 2017
E.g.f.: -(15/16) + (9/16)*x - x^2/16 + (1/16)*(15 +9*x +x^2)*exp(2*x). - G. C. Greubel, Jun 05 2023
MATHEMATICA
z = 60; s = x/(1-x); p = (1 - s^2)^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291013 *)
LinearRecurrence[{6, -12, 8}, {0, 3, 6, 15, 36, 85}, 41] (* G. C. Greubel, Jun 05 2023 *)
PROG
(PARI) concat(0, Vec(x*(3 -12*x +15*x^2 -6*x^3 +x^4)/(1-2*x)^3 + O(x^50))) \\ Colin Barker, Aug 23 2017
(Magma) [0, 3, 6] cat [2^(n-6)*(60 +17*n +n^2): n in [3..40]]; // G. C. Greubel, Jun 05 2023
(SageMath) [(2^(n-2)*(60 +17*n +n^2) -15*int(n==0) + 9*int(n==1))//16 for n in range(41)] # G. C. Greubel, Jun 05 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 23 2017
STATUS
approved