A012494 Expansion of e.g.f. arctan(sin(x)) (odd powers only).

1, -3, 45, -1743, 125625, -14554683, 2473184805, -579439207623, 179018972217585, -70518070842040563, 34495620120141463965, -20515677772241956573503, 14578232896601243652363945, -12198268199871431840616166443, 11871344562637111570703016357525

Offset: 0

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

• sign
• easy

Comments

arctan(sin(x)) = x - 3*x^3/3! + 45*x^5/5! - 1743*x^7/7! + 125625*x^9/9! + ....

Absolute values are coefficients in expansion of

arctanh(arcsinh(x)) = x + 3*x^3/3! + 45*x^5/5! + 1743*x^7/7! + ....

arccot(sin(x)) = Pi/2 - x + 3*x^3/3! - 45*x^5/5! + 1743*x^7/7! - ....

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..127

Crossrefs

Bisection of A003704, A013208.

Cf. A101923, A001209, A000364, A000281, A156134, A002437.

Cf. other sequences with a g.f. of the form cos(x)/(1 - k*sin^2(x)): A000364 (k=1), A001209 (k=1/2), A000281 (k=2), A156134 (k=3), A002437 (k=4).

Programs

Maple

a:= n-> (t-> t!*coeff(series(arctan(sin(x)), x, t+1), x, t))(2*n+1):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 16 2018

Mathematica

Drop[ Range[0, 25]! CoefficientList[ Series[ ArcTan[ Sin[x]], {x, 0, 25}], x], {1, 25, 2}] (* Or *)
f[n_] := n!Sum[(1 + (-1)^(n - 2k + 1))2^(1 - 2k)Sum[(-1)^((n + 1)/2 - j)Binomial[2k - 1, j]((2j - 2k + 1)^n/n!)/(2k - 1), {j, 0, (2k - 1)/2}], {k, Ceiling[n/2]}]; Table[ f[n], {n, 1, 25, 2}] (* Robert G. Wilson v *)

Maxima

a(n):=n!*sum((1+(-1)^(n-2*k+1))*2^(1-2*k)*sum((-1)^((n+1)/2-i)*binomial(2*k-1,i)*(2*i-2*k+1)^n/n!,i,0,(2*k-1)/2)/(2*k-1),k,1,ceiling((n)/2)); /* Vladimir Kruchinin, Feb 25 2011 */

Maxima

a(n):=sum(sum((2*i-2*k-1)^(2*n+1)*binomial(2*k+1,i)*(-1)^(n-i+1),i,0,k)/(4^k*(2*k+1)),k,0,n); /* Vladimir Kruchinin, Feb 04 2012 */

Formula

a(n) = n!*sum(k=1..ceiling(n/2), (1+(-1)^(n-2*k+1))*2^(1-2*k)*sum(i=0..(2*k-1)/2, (-1)^((n+1)/2-i)*binomial(2*k-1,i)*(2*i-2*k+1)^n/n!)/(2*k-1)), n>0. Vladimir Kruchinin, Feb 25 2011

G.f.: cos(x) /(1 + sin^2(x)) = 1 - 3*x^2/2! + 45*x^4/4! - ... . - Peter Bala, Feb 06 2017

a(n) ~ (-1)^n * (2*n)! / (log(1+sqrt(2)))^(2*n+1). - Vaclav Kotesovec, Aug 17 2018