%I #21 Jan 19 2021 14:13:43
%S 1,1,4,13,36,94,239,597,1471,3586,8669,20818,49726,118259,280239,
%T 662117,1560516,3670321,8617584,20203698,47308391,110659649,258614439,
%U 603929562,1409413761,3287385206,7664034874,17860302403
%N Expansion of -(4*x^3-7*x^2+4*x-1)/(2*x^4-5*x^3+8*x^2-5*x+1).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,5,-2).
%F a(n) = (n+1)*Sum_{k=0..n} (Sum_{i=0..n-k} (binomial(i+k,i)*2^i*binomial(2*k+2,n-i-k)*(-1)^(n-i-k))/(k+1)).
%F a(n) = 5*a(n-1)-8*a(n-2)+5*a(n-3)-2*a(n-4) for n>3, a(0)=1, a(1)=1, a(2)=4, a(3)=13.
%t Table[(n + 1) Sum[Sum[(Binomial[i + k, i] 2^i Binomial[2 k + 2, n - i - k] (-1)^(n - i - k))/(k + 1), {i, 0, n - k}], {k, 0, n}], {n, 0, 27}] (* or *)
%t CoefficientList[Series[-(4 x^3 - 7 x^2 + 4 x - 1)/(2 x^4 - 5 x^3 + 8 x^2 - 5 x + 1), {x, 0, 27}], x] (* _Michael De Vlieger_, Apr 01 2016 *)
%t LinearRecurrence[{5,-8,5,-2},{1,1,4,13},30] (* _Harvey P. Dale_, Jan 19 2021 *)
%o (Maxima) a(n):=(n+1)*sum(sum(binomial(i+k,i)*2^i*binomial(2*k+2,n-i-k)*(-1)^(n-i-k),i,0,n-k)/(k+1),k,0,n);
%o (PARI) x='x+O('x^99); Vec(-(4*x^3-7*x^2+4*x-1)/(2*x^4-5*x^3+8*x^2-5*x+1)) \\ _Altug Alkan_, Apr 01 2016
%Y Cf. A034008.
%K nonn,easy
%O 0,3
%A _Vladimir Kruchinin_, Apr 01 2016
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