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
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (9,-18).
Crossrefs
Programs
-
Magma
[2*6^n -3^n: n in [0..40]]; // G. C. Greubel, Nov 14 2024
-
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 *) -
PARI
Vec(1/(1-3*x)/(1-6*x)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012
-
Sage
[lucas_number1(n,9,18) for n in range(1,41)] # Zerinvary Lajos, Apr 23 2009
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
Extensions
More terms added by G. C. Greubel, Nov 14 2024