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A144902 Expansion of x/((1-x-x^3)*(1-x)^8).

%I A144902 #12 Jul 27 2022 10:36:28
%S A144902 0,1,9,45,166,505,1342,3224,7161,14938,29602,56211,102973,182963,
%T A144902 316694,535947,889454,1451305,2333356,3703510,5812615,9034001,
%U A144902 13921551,21294946,32364747,48915873,73576675,110213470,164508959,244810154,363371304,538175735
%N A144902 Expansion of x/((1-x-x^3)*(1-x)^8).
%H A144902 Vincenzo Librandi, <a href="/A144902/b144902.txt">Table of n, a(n) for n = 0..1000</a>
%H A144902 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,85,-134,154,-140,106,-65,29,-8,1).
%F A144902 G.f.: x/((1-x-x^3)*(1-x)^8).
%F A144902 From _G. C. Greubel_, Jul 27 2022: (Start)
%F A144902 a(n) = Sum_{j=0..floor((n+7)/3)} binomial(n-2*j+7, j+8).
%F A144902 a(n) = A099567(n+7, 8). (End)
%p A144902 a:= n-> (Matrix(11, (i, j)-> if i=j-1 then 1 elif j=1 then [9, -36, 85, -134, 154, -140, 106, -65, 29, -8, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
%t A144902 CoefficientList[Series[x/((1-x-x^3)(1-x)^8), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)
%o A144902 (Magma)
%o A144902 A144903:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+7, j+8): j in [0..Floor((n+7)/3)]]) >;
%o A144902 [A144903(n): n in [0..40]]; // _G. C. Greubel_, Jul 27 2022
%o A144902 (SageMath)
%o A144902 def A144903(n): return sum(binomial(n-2*j+7, j+8) for j in (0..((n+7)//3)))
%o A144902 [A144903(n) for n in (0..40)] # _G. C. Greubel_, Jul 27 2022
%Y A144902 9th column of A144903.
%Y A144902 Cf. A099567.
%K A144902 nonn,easy
%O A144902 0,3
%A A144902 _Alois P. Heinz_, Sep 24 2008