A075554 Denominators in the Maclaurin series for arctan(1+x).

2, 4, 12, 1, 40, 48, 112, 1, 288, 320, 704, 1, 1664, 1792, 3840, 1, 8704, 9216, 19456, 1, 43008, 45056, 94208, 1, 204800, 212992, 442368, 1, 950272, 983040, 2031616, 1, 4325376, 4456448, 9175040, 1, 19398656, 19922944, 40894464, 1, 85983232

Offset: 1

Author

Eric W. Weisstein, Sep 23 2002

Keywords

• nonn
• easy

Comments

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A075553, A132314.

Programs

Mathematica

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 *)

Maxima

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));
makelist(denom(atan(n),n,1,10); /* Vladimir Kruchinin, Apr 17 2014 */

Formula

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

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