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a(n) = n^4*Fibonacci(n).
2

%I #15 Aug 25 2024 18:49:09

%S 0,1,16,162,768,3125,10368,31213,86016,223074,550000,1303049,2985984,

%T 6654713,14482832,30881250,64684032,133383037,271257984,544872101,

%U 1082400000,2128789026,4148908016,8019403537,15383789568,29306640625,55473687568,104384578338

%N a(n) = n^4*Fibonacci(n).

%H Colin Barker, <a href="/A259547/b259547.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,-10,15,11,-15,-10,5,5,1).

%F G.f.: -x*(x^8-11*x^7+87*x^6-48*x^5+240*x^4+48*x^3+87*x^2+11*x+1) / (x^2+x-1)^5.

%F E.g.f.: exp(x/2)*x*(5*(1 + 7*x + 12*x^2 + 3*x^3)*cosh(sqrt(5)*x/2) + sqrt(5)*(1 + 21*x + 24*x^2 + 7*x^3)*sinh(sqrt(5)*x/2))/5. - _Stefano Spezia_, Aug 25 2024

%p a:= n-> n^4*(<<1|1>, <1|0>>^n)[1, 2]:

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jun 30 2015

%t Table[n^4 Fibonacci[n],{n,0,30}] (* or *) LinearRecurrence[{5,-5,-10,15,11,-15,-10,5,5,1},{0,1,16,162,768,3125,10368,31213,86016,223074},30] (* _Harvey P. Dale_, Mar 09 2016 *)

%o (PARI) a(n) = n^4*fibonacci(n)

%o (PARI) concat(0, Vec(-x*(x^8 -11*x^7 +87*x^6 -48*x^5 +240*x^4 +48*x^3 +87*x^2 +11*x +1)/(x^2 +x -1)^5 + O(x^50)))

%Y Cf. A000045, A000583, A045925, A259451, A259546.

%K nonn,easy

%O 0,3

%A _Colin Barker_, Jun 30 2015