A101923 Expansion of 2 * arccot(cos(x)).

1, 2, 1, -148, -3719, -20098, 5055961, 403644152, 7831409041, -2707151879398, -472143935754479, -22085804322342748, 9362259685093715401, 2995219209329323622102, 274269338931958691728681, -132963342779629343323496848, -70698673853383423350187244639

Offset: 1

Author

Ralf Stephan, Dec 27 2004

Keywords

• sign
• easy

Comments

Odd coefficients are zero.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A012494, A000364, A000464, A156138, A002439.

Cf. other sequences with a g.f. of the form sin(x)/(1 - k*sin^2(x)): A012494 (k=-1), A000364 (k=1), A000464 (k=2), A156138 (k=3), A002439 (k=4).

Programs

Maple

with(gfun):
series(sin(x)/(1-(1/2)*sin(x)^2), x, 35):
L := seriestolist(%):
seq(op(2*i, L)*(2*i-1)!, i = 1..floor((1/2)*nops(L)));
# Peter Bala, Feb 06 2017

Mathematica

With[{nn=40},Take[CoefficientList[Series[2ArcCot[Cos[x]],{x,0,nn}],x] Range[0,nn]!,{3,-1,2}]] (* Harvey P. Dale, Nov 17 2014 *) (* adapted by Vincenzo Librandi, Feb 07 2017 *)

Formula

2*acot(cos(x)) = Pi/2 + x^2/2! + 2*x^4/4! + x^6/6! - 148*x^8/8! - 3719*x^10/10! -...

2*atan(cos(x)) = Pi/2 - x^2/2! - 2*x^4/4! - x^6/6! + 148*x^8/8! + 3719*x^10/10! +...

G.f. sin(x)/(1 - 1/2*sin(x)^2) = x + 2*x^3/3! + x^5/5! - 148*x^7/7! - ... - Peter Bala, Feb 06 2017

Extensions

More terms from Harvey P. Dale, Nov 17 2014

Signs of the data entries corrected by Peter Bala, Feb 06 2017