A144898 - OEIS

%I #14 Jul 28 2022 09:12:04

%S 0,1,5,15,36,76,147,267,463,775,1262,2011,3150,4867,7438,11268,16951,

%T 25358,37766,56047,82945,122482,180553,265798,390880,574358,843432,

%U 1237966,1816384,2664311,3907237,5729077,8399372,12313154,18049371,26456513,38778103

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

%H Vincenzo Librandi, <a href="/A144898/b144898.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,11,-9,7,-4,1).

%F G.f.: x/((1-x-x^3)*(1-x)^4).

%F From _G. C. Greubel_, Jul 27 2022: (Start)

%F a(n) = Sum_{j=0..floor((n+3)/3)} binomial(n-2*j+3, j+4).

%F a(n) = A099567(n+3, 4). (End)

%p 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 CoefficientList[Series[ x/((1-x-x^3)(1-x)^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)

%o (Magma)

%o 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(n): n in [0..40]]; // _G. C. Greubel_, Jul 27 2022

%o (SageMath)

%o def A144898(n): return sum(binomial(n-2*j+3, j+4) for j in (0..((n+3)//3)))

%o [A144898(n) for n in (0..40)] # _G. C. Greubel_, Jul 27 2022

%Y 5th column of A144903.

%Y Cf. A000930, A050228, A077868, A144899, A144900, A144901, A144902, A144903, A144904, A226405.

%Y Cf. A078012, A099567, A135851.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Sep 24 2008