A012026 Expansion of e.g.f. tanh(sin(arctan(x))) = tanh(x/sqrt(1+x^2)) (odd powers only).

1, -5, 121, -6677, 651985, -98741477, 21378584137, -6259615830197, 2380487154907681, -1140222272527932869, 671453734616884934041, -476714184862146843059285, 401522701697360654591942641

Offset

1,2

Links

Table of n, a(n) for n=1..13.

Formula

a(n) = ((2*n+1)!*sum(m=0..n, binomial(n-1/2,n-m)/(2*m+1)!*sum(k=1..2*m+1, (-1)^(n-m+k+1)*k!*2^(2*m+1-k)*Stirling2(2*m+1,k)))). - Vladimir Kruchinin, Jun 17 2011

E.g.f.: tanh(x/sqrt(1+x^2)) = (x/sqrt(1+x^2))*G(0) where G(k)= 1 - x^2/(x^2 + (1+x^2)*(2*k+1)*(2*k+3)/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 06 2012

a(n) ~ (2*n-1)! * (-1)^(n+1) * 16 * (4+Pi^2)^(n-3/2) / Pi^(2*n). - Vaclav Kotesovec, Feb 02 2015

Example

tanh(sin(arctan(x))) = x - (5/3!)*x^3 + (121/5!)*x^5 - (6677/7!)*x^7 + (651985/9!)*x^9 - ...

Mathematica

nn = 20; Table[(CoefficientList[Series[Tanh[x/Sqrt[1 + x^2]], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* Vaclav Kotesovec, Feb 02 2015 *)

Prog

(Maxima)
a(n):=((2*n+1)!*sum(binomial(n-1/2, n-m)/(2*m+1)!*sum((-1)^(n-m+k+1)*k!*2^(2*m+1-k)*stirling2(2*m+1, k), k, 1, 2*m+1), m, 0, n)); /* Vladimir Kruchinin, Jun 17 2011 */

Crossrefs

Sequence in context: A282271 A179299 A012179 * A012190 A012077 A012046

Adjacent sequences: A012023 A012024 A012025 * A012027 A012028 A012029

Keyword

sign

Author

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

Status

approved