A075554 - OEIS

%I #22 Jun 24 2023 19:11:48

%S 2,4,12,1,40,48,112,1,288,320,704,1,1664,1792,3840,1,8704,9216,19456,

%T 1,43008,45056,94208,1,204800,212992,442368,1,950272,983040,2031616,1,

%U 4325376,4456448,9175040,1,19398656,19922944,40894464,1,85983232

%N Denominators in the Maclaurin series for arctan(1+x).

%C Terms with mod(n,4)=0 are zero, so a(n)=1 for those n.

%C arctan(1 + x) = Pi/4 + integral_{0..x} dt / (2 + 2*t + t^2). - _Michael Somos_, Apr 20 2014

%H Vincenzo Librandi, <a href="/A075554/b075554.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Denominator(sum(k=1..n, (sum(i=1..k, (2^(i-n-1)*(-1)^(i+n+(k-1)/2)/i*binomial(k-1,k-i))))*binomial(n-1,n-k))). - _Vladimir Kruchinin_, Apr 17 2014

%F Empirical g.f.: -x*(16*x^11 -16*x^10 -16*x^9 -24*x^8 -8*x^7 +4*x^6 +12*x^5 +22*x^4 +x^3 +12*x^2 +4*x +2) / ((x -1)*(x +1)*(x^2 +1)*(2*x^2 -1)^2*(2*x^2 +1)^2). - _Colin Barker_, Apr 18 2014

%t Table[Denominator[(-1)^n*2^(-n-1)*((1+I)^n-(1-I)^n)*I/n], {n, 1, 41}] (* _Jean-François Alcover_, Apr 18 2014, after _Vladimir Kruchinin_ *)

%o (Maxima)

%o atan(n):=(sum((sum((2^(i-n-1)*(-1)^(i+n+(k-1)/2)/i*binomial(k-1,k-i)),i,1,k))*binomial(n-1,n-k),k,1,n));

%o makelist(denom(atan(n),n,1,10); /* _Vladimir Kruchinin_, Apr 17 2014 */

%Y Cf. A075553, A132314.

%K nonn,easy

%O 1,1

%A _Eric W. Weisstein_, Sep 23 2002