A296788 Expansion of e.g.f. exp(x*arcsinh(x)) (even powers only).

1, 2, 8, 54, 104, 18810, -1648428, 247726374, -49445941200, 12841169289714, -4206667789245780, 1697448414191239710, -827415782970517712376, 479396168140498731959850, -325673237888367403728512700, 256401822876859593450127851030, -231597610351491427264049084814240

Offset

0,2

Links

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

Formula

a(n) = (2*n)! * [x^(2*n)] exp(x*arcsinh(x)).

a(n) ~ -(-1)^n * 2^(2*n) * n^(2*n-1) / exp(2*n + Pi/2). - Vaclav Kotesovec, Dec 21 2017

Example

exp(x*arcsinh(x)) = 1 + 2*x^2/2! + 8*x^4/4! + 54*x^6/6! + 104*x^8/8! + ...

Mathematica

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

Crossrefs

Cf. A001818, A006228, A009214, A009233, A079484, A259647, A296787, A296789.

Sequence in context: A197795 A327354 A197931 * A079503 A052694 A069729

Adjacent sequences: A296785 A296786 A296787 * A296789 A296790 A296791

Keyword

sign

Author

Ilya Gutkovskiy, Dec 20 2017

Status

approved