A291019 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3 + S^4.

1, 3, 9, 25, 68, 185, 504, 1373, 3739, 10180, 27714, 75445, 205376, 559064, 1521840, 4142609, 11276581, 30695881, 83556891, 227449066, 619135745, 1685339900, 4587637263, 12487934387, 33993205996, 92532358762, 251880840375, 685640764594, 1866371634554

Offset: 0

Author

Clark Kimberling, Aug 23 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 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 (5, -8, 6, -3)

Crossrefs

Cf. A000012, A289780, A291000.

Programs

Mathematica

z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3 + s^4;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291019 *)

Formula

G.f.: (1 - 2 x + 2 x^2 - 2 x^3)/(1 - 5 x + 8 x^2 - 6 x^3 + 3 x^4).

a(n) = 5*a(n-1) - 8*a(n-2) + 6*a(n-3) - 3*a(n-4) for n >= 5.