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Expansion of x*(3*x+1)*(1 - x^2 - x^3)/(1 - x - 15*x^2 - 19*x^3 + 20*x^4).
2

%I #22 Oct 27 2019 22:33:54

%S 1,4,18,93,416,2073,9720,46859,223726,1069831,5121642,24482721,

%T 117159620,560315013,2680448172,12821551727,61331067154,293376558067,

%U 1403343084750,6712850697141,32110530228584,153599278134609

%N Expansion of x*(3*x+1)*(1 - x^2 - x^3)/(1 - x - 15*x^2 - 19*x^3 + 20*x^4).

%C Sum of the top row elements of the n-th matrix power of the 9 X 9 matrix shown in the Mathematica program.

%H G. C. Greubel, <a href="/A123589/b123589.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1, 15, 19, -20).

%t M = {{0, 1, 1, 0, 1, 0, 0, 1, 0}, {1, 0, 1, 0, 0, 1, 0, 0, 1}, {1, 1, 0, 1, 1, 0, 1, 1, 0}, {0, 1, 0, 0, 1, 1, 0, 1, 0}, {0, 0, 1, 1, 0, 1, 0, 0, 1}, {1, 1, 0, 1, 1, 1, 1, 1, 0}, {0, 1, 0, 0, 1, 0, 0, 1, 1}, {0, 0, 1, 0, 0, 1, 1, 0, 1}, {1, 1, 0, 1, 1, 0, 1, 1, 0}}; v[1] = {1, 1, 1, 1, 1, 1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Floor[v[n][[1]]], {n, 1, 50}]

%t Rest[CoefficientList[Series[x*(3*x + 1)*(1 - x^2 - x^3)/(1 - x - 15*x^2 - 19*x^3 + 20*x^4), {x, 0, 50}], x]] (* _G. C. Greubel_, Oct 16 2017 *)

%o (PARI) Vec(x*(3*x+1)*(1-x^2-x^3)/(1-x-15*x^2-19*x^3+20*x^4)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012

%Y Cf. A120658.

%K nonn,easy

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Nov 12 2006