A291021 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3 + S^4 + S^5.

1, 3, 9, 25, 67, 178, 472, 1249, 3297, 8685, 22843, 60014, 157540, 413289, 1083693, 2840521, 7443331, 19500394, 51079696, 133782385, 350354841, 917456901, 2402365387, 6290338310, 16470047644, 43122600825, 112903347237, 295598625697, 773914899475

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 (6, -13, 14, -9, 2)

Crossrefs

Cf. A000012, A289780, A291000.

Programs

Mathematica

z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3 + s^4 + s^5;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291021 *)

Formula

G.f.: (-1 + 3 x - 4 x^2 + 4 x^3 - x^4)/(-1 + 6 x - 13 x^2 + 14 x^3 - 9 x^4 + 2 x^5).

a(n) = 6*a(n-1) - 13*a(n-2) + 14*a(n-3) - 9*a(n-4) + 2*a(n-5) for n >= 5.