A296463 Expansion of e.g.f. arcsinh(x)*arctanh(x) (even powers only).

0, 2, 4, 158, 3624, 427482, 29665260, 6948032310, 991515848400, 383952670412850, 93532380775766100, 53913667654307868750, 20087427376748637675000, 16096655588343149442026250, 8531309209053208518037597500, 9057367559484733295974741323750, 6486329752640392315697926589700000

Offset

0,2

Links

Table of n, a(n) for n=0..16.

Formula

E.g.f.: arcsin(x)*arctan(x) (even powers only, absolute values).

E.g.f.: (log(1 + x) - log(1 - x))*log(x + sqrt(1 + x^2))/2 (even powers only).

a(n) ~ (2*n-1)! * log(1+sqrt(2)) * (1 - (-1)^n * sqrt(Pi) / (4 * log(1+sqrt(2)) * sqrt(n))). - Vaclav Kotesovec, Dec 13 2017

Example

arcsinh(x)*arctanh(x) = 2*x^2/2! + 4*x^4/4! + 158*x^6/6! + 3624*x^8/8! + 427482*x^10/10! + ..

Mathematica

nmax = 16; Table[(CoefficientList[Series[ArcSinh[x] ArcTanh[x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[(Log[1 + x] - Log[1 - x]) Log[x + Sqrt[1 + x^2]]/2, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Crossrefs

Cf. A001818, A009744, A009747, A010050, A012752, A296462.

Sequence in context: A192063 A326796 A018558 * A132528 A018573 A018583

Adjacent sequences: A296460 A296461 A296462 * A296464 A296465 A296466

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 13 2017

Status

approved