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A296788 Expansion of e.g.f. exp(x*arcsinh(x)) (even powers only). 2

%I #7 Dec 21 2017 06:29:47

%S 1,2,8,54,104,18810,-1648428,247726374,-49445941200,12841169289714,

%T -4206667789245780,1697448414191239710,-827415782970517712376,

%U 479396168140498731959850,-325673237888367403728512700,256401822876859593450127851030,-231597610351491427264049084814240

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

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

%F a(n) ~ -(-1)^n * 2^(2*n) * n^(2*n-1) / exp(2*n + Pi/2). - _Vaclav Kotesovec_, Dec 21 2017

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

%t 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}]

%t 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}]

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

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Dec 20 2017

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