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 A180657 Numerator of the fraction tan( Sum_{k=1..n} arctan(k) ). 1

%I

%S 1,-3,0,4,-9,105,-308,36,-423,2387,-26004,104472,-50617,119889,

%T -5466072,3154072,51692571,-2921193,351666136,-1278405156,11188330461,

%U -68445012691,553299094188,-4915961459556

%N Numerator of the fraction tan( Sum_{k=1..n} arctan(k) ).

%H T. Amdeberhan, Luis A. Medina and V. H. Moll, <a href="http://dx.doi.org/10.1016/j.jnt.2007.05.008">Arithmetical properties of a sequence arising from an arctangent sum</a>, J. Number Theory 128 (2008) 1807-1846.

%H V. H. Moll, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.816.8525">An arithmetic conjecture on a sequence of arctangent sums</a>, 2012. - From _N. J. A. Sloane_, Dec 22 2012

%F Let x(1)=1 and x(n) = (x(n-1)+n)/(1-n*x(n-1)). Then a(n) = numerator(x(n)).

%e The fractions are x(1)=1, x(2)=-3, x(3)=0, x(4)=4, x(5)=-9/19, x(6)=105/73 etc.

%p A := proc(n) local x,itr; x := 1 ; for itr from 2 to n do x := (x+itr)/(1-itr*x) ; end do; numer(x) ; end proc:

%p seq(A(n),n=1..30) ;

%t x[1] := 1; x[n_] := (x[n - 1] + n)/(1 - n * x[n - 1]); Table[Numerator[x[n]], {n,10}] (* _Alonso del Arte_, Jan 21 2011 *)

%t (* Just for verification: *)

%t x[n_] := Tan[Sum[ArcTan[k], {k, n}]] // TrigExpand;

%t Table[x[n] // Numerator, {n, 24}] (* _Jean-François Alcover_, Mar 29 2020 *)

%Y For denominators see A220447.

%K sign,frac,easy,changed

%O 1,2

%A _R. J. Mathar_, Jan 21 2011

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Last modified April 3 01:31 EDT 2020. Contains 333195 sequences. (Running on oeis4.)