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%I #14 Sep 08 2022 08:45:17
%S 0,1,5,18,60,198,655,2171,7200,23880,79200,262669,871145,2889162,
%T 9581940,31778622,105394195,349541159,1159257600,3844692240,
%U 12750969600,42288749161,140251162205,465144722658,1542658254060,5116245273558
%N Expansion of x/(1-5*x+7*x^2-5*x^3+x^4).
%C Transform of the Fibonacci numbers under the Riordan array (1/(1-x)^2,x(1-x)^2)) (convolution array of natural numbers).
%H Harvey P. Dale, <a href="/A104630/b104630.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5, -7, 5, -1).
%F a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) - a(n-4).
%F a(n) = Sum_{k=0..n} binomial(n+k+1, 2*k+1)*F(k), where F(n) = Fibonacci(n).
%F a(n) = -a(-2-n) for all n in Z.
%e G.f. = x + 5*x^2 + 18*x^3 + 60*x^4 + 198*x^5 + 655*x^6 + 2171*x^7 + 7200*x^8 + ... - _Michael Somos_, Aug 12 2018
%t CoefficientList[Series[x/(1-5x+7x^2-5x^3+x^4),{x,0,30}],x] (* or *) LinearRecurrence[{5,-7,5,-1},{0,1,5,18},30] (* _Harvey P. Dale_, Sep 14 2013 *)
%t Table[Sum[Binomial[n+k+1,2*k+1]*Fibonacci[k], {k,0,n}], {n,0,50}] (* _G. C. Greubel_, Aug 12 2018 *)
%t a[ n_] := Sign[n + 1] SeriesCoefficient[ x / (1 - 5 x + 7 x^2 - 5 x^3 + x^4), {x, 0, Max[n, -2 - n]}]; (* _Michael Somos_, Aug 12 2018 *)
%o (PARI) x='x+O('x^50); concat([0], Vec(x/(1-5*x+7*x^2-5*x^3+x^4))) \\ _G. C. Greubel_, Aug 12 2018
%o (PARI) for(n=0,50, print1(sum(k=0,n, binomial(n+k+1, 2*k+1)*fibonacci(k)), ", ")) \\ _G. C. Greubel_, Aug 12 2018
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x/(1-5*x+7*x^2-5*x^3+x^4))); // _G. C. Greubel_, Aug 12 2018
%K easy,nonn
%O 0,3
%A _Paul Barry_, Mar 17 2005
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