A289795 p-INVERT of (3n), where p(S) = 1 - S - S^2.

3, 24, 162, 1083, 7260, 48681, 326406, 2188536, 14674041, 98388840, 659693103, 4423214952, 29657473194, 198852130383, 1333295304660, 8939689838877, 59940250397646, 401896898269128, 2694702070258437, 18067865859946320, 121144292846335179, 812267469938047224

Offset: 0

Author

Clark Kimberling, Aug 12 2017

Keywords

• nonn
• easy

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 A289780 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 (7, -3, 7, -1)

Crossrefs

Cf. A008585, A289780, A289796.

Programs

Mathematica

z = 60; s = 3*x/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A008585 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289795 *)
u/3  (* A289796 *)

Formula

G.f.: 3 (1 + x + x^2)/(1 - 7 x + 3 x^2 - 7 x^3 + x^4).

a(n) = 7*a(n-1) - 3*a(n-2) + 7*a(n-3) - a(n-4).