A296623 Expansion of e.g.f. log(1 + arctan(x)*arctanh(x)) (even powers only).
0, 2, -12, 448, -21728, 2380032, -318185472, 69695846400, -18235768762368, 6697099792220160, -2892199532135841792, 1606188416621920911360, -1034069421398404544593920, 810882197441673837894696960, -727447103613537543910242385920, 766865924510666637669136261447680
Offset: 0
Keywords
Examples
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! + ...
Programs
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Mathematica
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}] 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}]
Formula
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).