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%I #16 Jul 27 2022 10:37:13
%S 0,1,7,28,85,218,498,1045,2055,3840,6887,11945,20153,33228,53741,
%T 85522,134254,208344,320200,488103,738951,1112281,1666164,2485845,
%U 3696406,5481325,8109676,11975993,17658694,26005706,38259955,56243281,82625979,121321831,178067054
%N Expansion of x/((1-x-x^3)*(1-x)^6).
%H Vincenzo Librandi, <a href="/A144900/b144900.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,36,-41,36,-27,16,-6,1).
%F G.f.: x/((1-x-x^3)*(1-x)^6).
%F From _G. C. Greubel_, Jul 27 2022: (Start)
%F a(n) = Sum_{j=0..floor((n+5)/3)} binomial(n-2*j+5, j+6).
%F a(n) = A099567(n+5, 6). (End)
%p a:= n-> (Matrix(9, (i, j)-> if i=j-1 then 1 elif j=1 then [7, -21, 36, -41, 36, -27, 16, -6, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
%t CoefficientList[Series[x/((1-x-x^3)(1-x)^6), {x,0,40}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)
%t LinearRecurrence[{7,-21,36,-41,36,-27,16,-6,1},{0,1,7,28,85,218,498,1045,2055},40] (* _Harvey P. Dale_, Mar 02 2016 *)
%o (Magma)
%o A144900:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+5, j+6): j in [0..Floor((n+5)/3)]]) >;
%o [A144900(n): n in [0..40]]; // _G. C. Greubel_, Jul 27 2022
%o (SageMath)
%o def A144900(n): return sum(binomial(n-2*j+5, j+6) for j in (0..((n+5)//3)))
%o [A144900(n) for n in (0..40)] # _G. C. Greubel_, Jul 27 2022
%Y 7th column of A144903.
%Y Cf. A099567.
%K nonn,easy
%O 0,3
%A _Alois P. Heinz_, Sep 24 2008
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