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%I #36 Sep 28 2023 02:12:30
%S 1,10,69,410,2261,11970,61909,315850,1598421,8050130,40425749,
%T 202656090,1014866581,5079099490,25409813589,127092049130,
%U 635589254741,3178333432050,15892828897429,79467630222970,397348609370901,1986774423719810,9933966253389269
%N Expansion of g.f. 1/((1-2x)(1-3x)(1-5x)).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (10,-31,30).
%F From _Hieronymus Fischer_, Jun 25 2007: (Start)
%F a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*5^k.
%F a(n) = (2^(n+3) + 5^(n+2) - 3^(n+3))/6. (End)
%F a(n) = ((5^n - 2^n)/3 - (3^n - 2^n))/2 , n >= 2. - _Zerinvary Lajos_, Jun 05 2009
%F From _Vincenzo Librandi_, Mar 15 2011: (Start)
%F a(n) = 10*a(n-1) - 31*a(n-2) + 30*a(n-3), n >= 3.
%F a(n) = 8*a(n-1) - 15*a(n-2) + 2^n, a(0)=1, a(1)=10. (End)
%F E.g.f.: exp(2*x)*(8 - 27*exp(x) + 25*exp(3*x))/6. - _Stefano Spezia_, Sep 27 2023
%t CoefficientList[ Series[ 1/((1 - 2x)(1 - 3x)(1 - 5x)), {x, 0, 20} ], x ]
%t LinearRecurrence[{10,-31,30},{1,10,69},20] (* _Harvey P. Dale_, Oct 05 2014 *)
%o (Sage) [((5^n - 2^n)/3-(3^n - 2^n))/2 for n in range(2,22)] # _Zerinvary Lajos_, Jun 05 2009
%o (PARI) Vec(1/((1-2*x)*(1-3*x)*(1-5*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y Column k=3 of A343751.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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