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A016137

Expansion of 1/((1-3*x)*(1-6*x)).

1, 9, 63, 405, 2511, 15309, 92583, 557685, 3352671, 20135709, 120873303, 725416965, 4353033231, 26119793709, 156723545223, 940355620245, 5642176768191, 33853189749309, 203119525916343, 1218718317759525, 7312313393341551, 43873890820402509, 263243376303474663, 1579460351964026805, 9476762394213697311

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OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-18).

FORMULA

a(n) = (3^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).

a(n) = 2*6^n - 3^n.

E.g.f.: (d^2/dx^2)((((exp(3*x)-1)/3)^2)/2!) = -exp(3*x) + 2*exp(6*x).

With leading zero, this is (6^n - 3^n)/3, the binomial transform of A016127 (with extra leading zero). - Paul Barry, Aug 20 2003

With leading zero, this is the fourth binomial transform of A001045, with a(n) = (2^n-1)(3^n/3 - 0^n/3) = A000225(n)*(A000244(n-1) - 0^n/3). - Paul Barry, Apr 28 2004

a(n) = Sum_{k=0..n} A100852(n,k). - Reinhard Zumkeller, Nov 20 2004

Sum_{k=1..n} 3^(k-1)*3^(n-k)*binomial(n, k). - Zerinvary Lajos, Sep 24 2006

a(n) = 9*a(n-1) - 18*a(n-2), n >= 2. - Vincenzo Librandi, Mar 14 2011

MATHEMATICA

Table[2*6^n -3^n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)

CoefficientList[Series[1/((1-3x)(1-6x)), {x, 0, 40}], x] (* or *) LinearRecurrence[{9, -18}, {1, 9}, 40] (* Harvey P. Dale, Jul 07 2012 *)

PROG

(SageMath) [lucas_number1(n, 9, 18) for n in range(1, 41)] # Zerinvary Lajos, Apr 23 2009

(PARI) Vec(1/(1-3*x)/(1-6*x)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

(Magma) [2*6^n -3^n: n in [0..40]]; // G. C. Greubel, Nov 14 2024

CROSSREFS

Second column of triangle A075498.