A305200 -id:A305200 - cp's OEIS frontend

A191719 Expansion of e.g.f. arctan(x*exp(x)).

0, 1, 2, 1, -20, -151, -354, 6217, 100472, 537777, -7631270, -223395919, -2120164188, 22050300505, 1154262915638, 17130776734905, -105423782758544, -11372993234072863, -245877012220234446, 345837436238423521, 188329590656514108380

Offset: 0

Vladimir Kruchinin, Jun 13 2011

sign

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A009635, A216401, A297009.

Rest[CoefficientList[Series[ArcTan[x*Exp[x]],{x,0,20}],x]*Range[0,20]!] (* Vaclav Kotesovec, Jan 02 2014 *)

a(n):=n!*sum(((2*m-1)^(n-2*m)*(-1)^(m-1))/(n-2*m+1)!,m,1,(n+1)/2);

a(n) = n!*Sum_{m=1..(n+1)/2} ((2*m-1)^(n-2*m)*(-1)^(m-1))/(n-2*m+1)!.

a(n) ~ (n-1)! * sin(n*arctan(1/tan(r))) * (cos(r)/r)^n, where r = Im(LambertW(I)) = A305200 = 0.576412723031435283148289239887... is the root of the equation exp(r*tan(r))=cos(r)/r. - Vaclav Kotesovec, Jan 02 2014

a(0)=0 prepended by Seiichi Manyama, Oct 01 2021

A305202 Decimal expansion of the imaginary part of continued exponential i.

Original entry on oeis.org

3, 7, 4, 6, 9, 9, 0, 2, 0, 7, 3, 7, 1, 1, 7, 4, 9, 3, 6, 0, 5, 9, 7, 8, 4, 2, 8, 7, 5, 9, 7, 2, 0, 8, 0, 7, 5, 1, 2, 8, 0, 2, 1, 7, 5, 3, 2, 6, 7, 8, 2, 6, 4, 2, 5, 5, 7, 5, 0, 2, 4, 3, 2, 5, 9, 1, 2, 2, 1, 5, 3, 1, 6, 5, 4, 9, 6, 7, 8, 1, 1, 6, 6, 4, 9, 6, 3, 6, 9, 8, 3, 4, 3, 7, 9, 1, 2, 7, 6, 6, 4, 5, 7, 0, 5

Offset: 0

Keerthi Vasan Gopala, May 27 2018

This is the imaginary part of e^(i*e^(i*e^(i...))).

			0.3746990207371174936059784287597208075128...

Eric Weisstein's World of Mathematics, Lambert W-Function
Wikipedia, Lambert W function

Cf. A305200.

RealDigits[Re[LambertW[I]], 10, 120][[1]] (* Vaclav Kotesovec, Oct 02 2021 *)

Equals Im(i*LambertW(-i)). - Alois P. Heinz, May 27 2018

Equals Re(LambertW(i)). - Vaclav Kotesovec, Oct 02 2021

More digits from Alois P. Heinz, May 27 2018

A305208 Decimal expansion of the real part of the continued exponential i/Pi.

Original entry on oeis.org

8, 8, 5, 3, 0, 2, 9, 2, 2, 6, 3, 1, 7, 2, 0, 6, 0, 1, 7, 3, 5, 6, 1, 1, 1, 6, 2, 3, 4, 1, 0, 6, 4, 9, 9, 5, 1, 8, 9, 5, 7, 7, 5, 3, 3, 9, 7, 9, 6, 7, 0, 9, 8, 4, 2, 1, 2, 1, 5, 3, 2, 7, 3, 0, 4, 4, 1, 4, 0, 4, 3, 1, 4, 8, 2, 6, 3, 9, 0, 4, 6, 3, 8, 2, 1, 5, 3, 8, 2, 2, 8, 5, 4, 0, 9, 2, 3, 7, 3, 1, 9, 0, 1, 1, 7, 8

Offset: 0

Keerthi Vasan Gopala, May 27 2018

This is the real part of e^((i/Pi)*e^((i/Pi)*e^((i/Pi)...))).

			0.88530292263172060173561116234106499518957753397967...

Cf. A049006, A077589, A077590, A305200, A305202, A305210.

Mathematica
```
Re[Pi*I*N[ProductLog[-I/Pi], 100]]
```

Equals Re(Pi*i*LambertW(-i/Pi)).

A305210 Decimal expansion of the imaginary part of continued exponential (i/Pi).

Original entry on oeis.org

2, 5, 6, 2, 9, 9, 5, 3, 7, 1, 6, 3, 8, 6, 1, 3, 1, 2, 5, 2, 9, 9, 9, 6, 7, 2, 9, 8, 8, 0, 9, 8, 2, 5, 3, 8, 0, 7, 8, 3, 4, 1, 4, 6, 3, 8, 8, 4, 0, 1, 4, 2, 1, 3, 3, 7, 7, 5, 1, 8, 9, 5, 0, 9, 9, 3, 7, 4, 1, 7, 4, 5, 1, 0, 9, 3, 3, 0, 9, 7, 5, 4, 9, 5, 2, 7, 6, 9, 1, 4, 7, 3, 7, 1, 0, 8, 2, 9, 4, 3, 6, 1, 3, 4

Offset: 0

Keerthi Vasan Gopala, May 27 2018

This is the imaginary part of e^((i/Pi)*e^((i/Pi)*e^((i/Pi)...))).

			0.256299537163861312529996729880982538078341463884...

Cf. A049006, A077589, A077590, A305200, A305202, A305208.

Mathematica
```
Im[Pi*I*N[ProductLog[-I/Pi], 100]]
```

Equals Im(Pi*i*LambertW(-i/Pi)).

cp's OEIS Frontend

A191719 Expansion of e.g.f. arctan(x*exp(x)).

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A305202 Decimal expansion of the imaginary part of continued exponential i.

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A305208 Decimal expansion of the real part of the continued exponential i/Pi.

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A305210 Decimal expansion of the imaginary part of continued exponential (i/Pi).

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