A296788 -id:A296788 - 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

A296789 Expansion of e.g.f. exp(x*arctanh(x)) (even powers only).

Original entry on oeis.org

1, 2, 20, 504, 24464, 1959840, 234852672, 39370660224, 8799246209280, 2528787321598464, 908585701684024320, 399070678264750356480, 210373049449102957645824, 131083661069772517440921600, 95304505860052894815543705600, 79961055068441273887848131297280

Offset: 0

Ilya Gutkovskiy, Dec 20 2017

nonn

			exp(x*arctanh(x)) = 1 + 2*x^2/2! + 20*x^4/4! + 504*x^6/6! + 24464*x^8/8! + ...

Cf. A000246, A009252, A009273, A010050, A166356, A259647, A293193, A296787, A296788.

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

a(n) ~ 2^(2*n + 2) * n^(2*n) / exp(2*n). - 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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