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

1,2

Comments

Odd coefficients are zero.

Links

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

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

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 *)

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).

Sequence in context: A081708 A358858 A012004 * A010788 A258819 A337051

Adjacent sequences: A101920 A101921 A101922 * A101924 A101925 A101926

Keyword

sign,easy

Author

Ralf Stephan, Dec 27 2004

Extensions

More terms from Harvey P. Dale, Nov 17 2014

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

Status

approved