%I #5 Dec 18 2017 11:54:40

%S 0,2,-12,448,-21728,2380032,-318185472,69695846400,-18235768762368,

%T 6697099792220160,-2892199532135841792,1606188416621920911360,

%U -1034069421398404544593920,810882197441673837894696960,-727447103613537543910242385920,766865924510666637669136261447680

%N Expansion of e.g.f. log(1 + arctan(x)*arctanh(x)) (even powers only).

%F E.g.f.: log(1 + (i/4)*(log(1 - i*x) - log(1 + i*x))*(log(1 + x) - log(1 - x))), where i is the imaginary unit (even powers only).

%e log(1 + arctan(x)*arctanh(x)) = 2*x^2/2! - 12*x^4/4! + 448*x^6/6! - 21728*x^8/8! + 2380032*x^10/10! - 318185472*x^12/12! + ...

%t nmax = 15; Table[(CoefficientList[Series[Log[1 + ArcTan[x] ArcTanh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

%t nmax = 15; Table[(CoefficientList[Series[Log[1 + (I/4) (Log[1 - I x] - Log[1 + I x]) (Log[1 + x] - Log[1 - x])], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

%Y Cf. A009272, A009398, A010050, A012727, A110708, A202139.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Dec 17 2017