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%I #34 Dec 07 2019 12:18:20
%S 1,9,63,405,2511,15309,92583,557685,3352671,20135709,120873303,
%T 725416965,4353033231,26119793709,156723545223,940355620245,
%U 5642176768191,33853189749309,203119525916343,1218718317759525
%N Expansion of 1/((1-3x)(1-6x)).
%C a(n) is the sum of n-th row in triangle A100852. - _Reinhard Zumkeller_, Nov 20 2004
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-18)
%F a(n) = (3^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).
%F a(n) = -3^n + 2*6^n.
%F E.g.f.: (d^2/dx^2)((((exp(3*x)-1)/3)^2)/2!) = -exp(3*x) + 2*exp(6*x).
%F With leading zero, this is (6^n - 3^n)/3, the binomial transform of A016127 (with extra leading zero). - _Paul Barry_, Aug 20 2003
%F 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
%F Sum_{k=1..n} 3^(k-1)*3^(n-k)*binomial(n, k). - _Zerinvary Lajos_, Sep 24 2006
%F a(n) = 9*a(n-1) - 18*a(n-2), n >= 2. - _Vincenzo Librandi_, Mar 14 2011
%t Join[{a=1,b=9},Table[c=9*b-18*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 27 2011 *)
%t CoefficientList[Series[1/((1-3x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{9,-18},{1,9},30] (* _Harvey P. Dale_, Jul 07 2012 *)
%o (Sage) [lucas_number1(n,9,18) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o (PARI) Vec(1/(1-3*x)/(1-6*x)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 24 2012
%Y Second column of triangle A075498.
%Y Cf. A017933.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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