A099138 a(n) = 6^(n-1)*J(n), where J(n) = A001045(n).

0, 1, 6, 108, 1080, 14256, 163296, 2006208, 23794560, 287214336, 3436494336, 41298398208, 495217981440, 5944792559616, 71324450021376, 855971764420608, 10271190988062720, 123257112966660096, 1479068428940476416

Offset: 0

Author

Paul Barry, Sep 29 2004

Keywords

• easy
• nonn

Comments

In general k^(n-1)*J(n), where J(n) = A001045(n), is given by ((2*k)^n - (-k)^n)/(3*k) with g.f. x/((1+k*x)*(1-2*k*x)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..925
• Index entries for linear recurrences with constant coefficients, signature (6,72).

Crossrefs

Cf. A001045, A003683, A080424, A091903, A091904.

Programs

Magma

[(12^n - (-6)^n)/18: n in [0..40]]; // G. C. Greubel, Feb 18 2023

Mathematica

LinearRecurrence[{6,72}, {0,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

SageMath

[(12^n - (-6)^n)/18 for n in range(41)] # G. C. Greubel, Feb 18 2023

Formula

G.f.: x/((1+6*x)*(1-12*x)).

a(n) = 6^(n-1)*Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k) * 2^k.

a(n) = (12^n - (-6)^n)/18.

a(n) = 6^(n-1)*A001045(n).

E.g.f.: (1/18)*(exp(12*x) - exp(-6*x)). - G. C. Greubel, Feb 18 2023