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Expansion of x/((1-x-x^3)*(1-x)^7).
8

%I #12 Jul 27 2022 10:36:32

%S 0,1,8,36,121,339,837,1882,3937,7777,14664,26609,46762,79990,133731,

%T 219253,353507,561851,882051,1370154,2109105,3221386,4887550,7373395,

%U 11069801,16551126,24660802,36636795,54295489,80301195,118561150,174804431,257430410

%N Expansion of x/((1-x-x^3)*(1-x)^7).

%H Vincenzo Librandi, <a href="/A144901/b144901.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,57,-77,77,-63,43,-22,7,-1).

%F G.f.: x/((1-x-x^3)*(1-x)^7).

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

%F a(n) = Sum_{j=0..floor((n+6)/3)} binomial(n-2*j+6, j+7).

%F a(n) = A099567(n+6, 7). (End)

%p a:= n-> (Matrix(10, (i, j)-> if i=j-1 then 1 elif j=1 then [8, -28, 57, -77, 77, -63, 43, -22, 7, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

%t CoefficientList[Series[x/((1-x-x^3)(1-x)^7), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)

%o (Magma)

%o A144901:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+6, j+7): j in [0..Floor((n+6)/3)]]) >;

%o [A144901(n): n in [0..40]]; // _G. C. Greubel_, Jul 27 2022

%o (SageMath)

%o def A144901(n): return sum(binomial(n-2*j+6, j+7) for j in (0..((n+6)//3)))

%o [A144901(n) for n in (0..40)] # _G. C. Greubel_, Jul 27 2022

%Y 8th column of A144903.

%Y Cf. A099567.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Sep 24 2008