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A226405 Expansion of x/((1-x-x^3)*(1-x)^3).

%I A226405 #18 Jul 28 2022 09:11:50
%S A226405 0,1,4,10,21,40,71,120,196,312,487,749,1139,1717,2571,3830,5683,8407,
%T A226405 12408,18281,26898,39537,58071,85245,125082,183478,269074,394534,
%U A226405 578418,847927,1242926,1821840,2670295,3913782,5736217,8407142,12321590,18058510,26466393
%N A226405 Expansion of x/((1-x-x^3)*(1-x)^3).
%C A226405 From _Bruno Berselli_, Jun 07 2013: (Start)
%C A226405 A050228(n)  = a(n)   -a(n-1), n>0.
%C A226405 A077868(n-1)= a(n) -2*a(n-1)   +a(n-2), n>1.
%C A226405 A000217(n)  = a(n)   -a(n-1)             -a(n-3), n>2.
%C A226405 A000930(n-1)= a(n) -3*a(n-1) +3*a(n-2)   -a(n-3), n>2.
%C A226405 n           = a(n) -2*a(n-1)   +a(n-2)   -a(n-3)   +a(n-4), n>3.
%C A226405 1           = a(n) -3*a(n-1) +3*a(n-2) -2*a(n-3) +2*a(n-4)   -a(n-5), n>4.
%C A226405 0           = a(n) -4*a(n-1) +6*a(n-2) -5*a(n-3) +4*a(n-4) -3*a(n-5) +a(n-6), n>5.
%C A226405 (End)
%H A226405 Alois P. Heinz, <a href="/A226405/b226405.txt">Table of n, a(n) for n = 0..1000</a>
%H A226405 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,5,-4,3,-1).
%F A226405 G.f.: x/((1-x-x^3)*(1-x)^3).
%F A226405 From _G. C. Greubel_, Jul 27 2022: (Start)
%F A226405 a(n) = Sum_{j=0..floor((n+2)/3)} binomial(n-2*j+2, j+3).
%F A226405 a(n) = A099567(n+2, 3). (End)
%p A226405 a:= n-> (Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -6, 5, -4, 3, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
%t A226405 LinearRecurrence[{4,-6,5,-4,3,-1}, {0,1,4,10,21,40}, 40]  (* _Bruno Berselli_, Jun 07 2013 *)
%t A226405 CoefficientList[Series[x/((1-x-x^3)*(1-x)^3), {x, 0, 50}], x] (* _G. C. Greubel_, Apr 28 2017 *)
%o A226405 (PARI) my(x='x+O('x^50)); Vec(x/((1-x-x^3)*(1-x)^3)) \\ _G. C. Greubel_, Apr 28 2017
%o A226405 (Magma)
%o A226405 A226405:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+2, j+3): j in [0..Floor((n+2)/3)]]) >;
%o A226405 [A226405(n): n in [0..40]]; // _G. C. Greubel_, Jul 27 2022
%o A226405 (SageMath)
%o A226405 def A226405(n): return sum(binomial(n-2*j+2, j+3) for j in (0..((n+2)//3)))
%o A226405 [A226405(n) for n in (0..40)] # _G. C. Greubel_, Jul 27 2022
%Y A226405 4th column of A144903.
%Y A226405 Cf. A000217, A078012, A099567, A135851.
%Y A226405 Cf. A000930, A050228, A077868, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.
%K A226405 nonn,easy
%O A226405 0,3
%A A226405 _Alois P. Heinz_, Jun 06 2013