A296789 -id:A296789 - cp's OEIS frontend

A296787 Expansion of e.g.f. exp(x*arctan(x)) (even powers only).

1, 2, 4, 24, -496, 36000, -3753408, 556961664, -111591202560, 29054584410624, -9541382573767680, 3858875286730168320, -1884995591107521540096, 1094305223336273239449600, -744771228363250138965196800, 587358379156469629707528929280

Offset: 0

Ilya Gutkovskiy, Dec 20 2017

sign

			exp(x*arctan(x)) = 1 + 2*x^2/2! + 4*x^4/4! + 24*x^6/6! - 496*x^8/8! + ...

Cf. A002019, A009252, A009273, A010050, A102059, A166356, A259647, A293192, A296788, A296789.

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

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

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

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

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 20 2017

sign

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

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

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}]

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

cp's OEIS Frontend

A296787 Expansion of e.g.f. exp(x*arctan(x)) (even powers only).

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