A009710 E.g.f. tan(tan(x)^2) (even powers only).

0, 2, 16, 512, 34816, 3821312, 618121216, 138682959872, 41171702972416, 15610723195092992, 7357121913006063616, 4217775794187229724672, 2889975739296119171055616, 2332177121915783600628826112, 2189206131249303252650023714816, 2365042784256046151642765393395712

Offset

0,2

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = 4 * Sum_{m=0..n-1/2} ((Sum_{j=1..2*m+1} j!*2^(2*m-j)*(-1)^(m+1+j)*Stirling2(2*m+1,j)) * (Sum_{j=4*m+2..2*n} binomial(j-1,4*m+1)*j!*2^(2*n-j-1)*(-1)^(n+1+j)*Stirling2(2*n,j)) / (2*m+1)!). - Vladimir Kruchinin, Jun 21 2011

a(n) ~ (2*n)! * 2 * sqrt(2) / ((2+Pi) * sqrt(Pi) * arctan(sqrt(Pi/2))^(2*n+1)). - Vaclav Kotesovec, Jan 24 2015

a(n) = 2*A024290(n). - Alois P. Heinz, Jan 04 2025

Example

tan(tan(x)*tan(x)) = 2/2!*x^2 + 16/4!*x^4 + 512/6!*x^6 + 34816/8!*x^8...

Mathematica

nn = 20; Table[(CoefficientList[Series[Tan[Tan[x]^2], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Jan 24 2015 *)

Prog

(Maxima) a(n):=4*sum(((sum(j!*2^(2*m-j)*(-1)^(m+1+j)*stirling2(2*m+1, j), j, 1, 2*m+1))*sum(binomial(j-1, 4*m+1)*j!*2^(2*n-j-1)*(-1)^(n+1+j)*stirling2(2*n, j), j, 4*m+2, 2*n))/(2*m+1)!, m, 0, n-1/2); /* Vladimir Kruchinin, Jun 21 2011 */

Crossrefs

Cf. A024290.

Sequence in context: A366513 A366517 A012389 * A012393 A189899 A189335

Adjacent sequences: A009707 A009708 A009709 * A009711 A009712 A009713

Keyword

nonn

Author

R. H. Hardin

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997

Status

approved