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A144898 Expansion of x/((1-x-x^3)*(1-x)^4).

%I A144898 #14 Jul 28 2022 09:12:04
%S A144898 0,1,5,15,36,76,147,267,463,775,1262,2011,3150,4867,7438,11268,16951,
%T A144898 25358,37766,56047,82945,122482,180553,265798,390880,574358,843432,
%U A144898 1237966,1816384,2664311,3907237,5729077,8399372,12313154,18049371,26456513,38778103
%N A144898 Expansion of x/((1-x-x^3)*(1-x)^4).
%H A144898 Vincenzo Librandi, <a href="/A144898/b144898.txt">Table of n, a(n) for n = 0..1000</a>
%H A144898 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,11,-9,7,-4,1).
%F A144898 G.f.: x/((1-x-x^3)*(1-x)^4).
%F A144898 From _G. C. Greubel_, Jul 27 2022: (Start)
%F A144898 a(n) = Sum_{j=0..floor((n+3)/3)} binomial(n-2*j+3, j+4).
%F A144898 a(n) = A099567(n+3, 4). (End)
%p A144898 a:= n-> (Matrix(7, (i, j)-> if i=j-1 then 1 elif j=1 then [5, -10, 11, -9, 7, -4, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
%t A144898 CoefficientList[Series[ x/((1-x-x^3)(1-x)^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)
%o A144898 (Magma)
%o A144898 A144898:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+3, j+4): j in [0..Floor((n+3)/3)]]) >;
%o A144898 [A144898(n): n in [0..40]]; // _G. C. Greubel_, Jul 27 2022
%o A144898 (SageMath)
%o A144898 def A144898(n): return sum(binomial(n-2*j+3, j+4) for j in (0..((n+3)//3)))
%o A144898 [A144898(n) for n in (0..40)] # _G. C. Greubel_, Jul 27 2022
%Y A144898 5th column of A144903.
%Y A144898 Cf. A000930, A050228, A077868, A144899, A144900, A144901, A144902, A144903, A144904, A226405.
%Y A144898 Cf. A078012, A099567, A135851.
%K A144898 nonn,easy
%O A144898 0,3
%A A144898 _Alois P. Heinz_, Sep 24 2008