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%I #26 Sep 08 2022 08:46:16
%S 0,1,1,2,7,10,20,34,65,124,230,430,800,1494,2792,5210,9727,18155,
%T 33892,63271,118110,220484,411588,768337,1434304,2677500,4998252,
%U 9330536,17417876,32515004,60697720,113308101,211519073,394855430,737100483,1375989990
%N Expansion of (x+4*x^4)/(1-x-x^2-x^4-2*x^5-x^8).
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,1,2,0,0,1).
%F a(n) = n*Sum_{j=0..(n-1)/3} binomial(n-3*j,n-4*j)*F(n-3*j)/(n-3*j), where F(n) = A000045(n).
%t CoefficientList[Series[(x + 4 x^4)/(1 - x - x^2 - x^4 - 2 x^5 - x^8), {x, 0, 35}], x] (* _Michael De Vlieger_, Mar 28 2016 *)
%t LinearRecurrence[{1,1,0,1,2,0,0,1},{0,1,1,2,7,10,20,34},40] (* _Harvey P. Dale_, Jul 03 2017 *)
%o (Maxima) a(n):=n*sum(binomial(n-3*j,n-4*j)/(n-3*j)*fib(n-3*j),j,0,(n-1)/3); /* or */ taylor((x+4*x^4)/(1-x-x^2-x^4-2*x^5-x^8),x,0,10);
%o (PARI) a(n) = n*sum(k=0, (n-1)/3, binomial(n-3*k,n-4*k)/(n-3*k)*fibonacci(n-3*k)); \\ _Altug Alkan_, Mar 25 2016
%o (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((x+4*x^4)/(1-x-x^2-x^4-2*x^5-x^8)));
%Y Cf. A000045.
%K nonn,easy
%O 0,4
%A _Vladimir Kruchinin_, Mar 25 2016
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