Mysterious connection

[mart_r]

Hello again,

some time ago I noticed that there seems to be a strange connection between the standard normal distribution and 1/tanh(pi^2), as to be seen in A133658 in the OEIS: http://www.research.att.com/~njas/sequences/A133658

I wondered if any of you are able to tell me what's going on with these numbers.

[mart_r]

Is this even interesting? A coincidence? Or is it a trivial phenomenon?

[10metreh]

Originally Posted by mart_r

Is this even interesting? A coincidence? Or is it a trivial phenomenon?

It's like that one where e^(pi*sqrt(163)) is nearly an integer.

[mart_r]

Had a look into this again today and found:

1/tanh(pi²) = 1 + 2*[e^(-2pi²)+e^(-4pi²)+e^(-6pi²)+e^(-8pi²)+...] (rather easy)
A133658 __= 1 + 2*[e^(-2pi²)+e^(-8pi²)+e^(-18pi²)+e^(-32pi²)+e^(-50pi²)+...] (rather interesting)

I would try to figure out why this is so, if only I could find an explanation of how the factor of 1/sqrt(2pi) in the standard normal distribution comes about...

(Come think of it, this thread should probably be moved to Math or MiscMath)

[davieddy]

Originally Posted by mart_r

I would try to figure out why this is so, if only I could find an explanation of how the factor of 1/sqrt(2pi) in the standard normal distribution comes about...

Are you familiar with the neat trick to find the area under the whole
curve e^(-ax^2)?

[mart_r]

Originally Posted by davieddy

Are you familiar with the neat trick to find the area under the whole
curve e^(-ax^2)?

Nope. Not yet.

[davieddy]

Hint:
Find the volume under the surface e^-a(x^2 + y^2)

Don't spoil the fun by googling

[mart_r]

Assuming a sphere (which I guess is the wrong way, right?),

e^(-3a(x²+y²)/2)/6pi^(3/2)

(is there something like an instruction for use of the TEX-format?)

[davieddy]

Originally Posted by davieddy

Hint:
Find the volume under the surface e^-a(x^2 + y^2)

Don't spoil the fun by googling

I am also frustrated by my inability to script maths
as I do on paper.

z = e^-a(x^2 + y^2) is the surface
Find the the volume between it and the whole plane z=0.

PS Pythagoras comes in handy.

[TimSorbet]

http://www.mersenneforum.org/showthread.php?t=4576

[mart_r]

Originally Posted by davieddy

z = e^-a(x^2 + y^2) is the surface
Find the the volume between it and the whole plane z=0.

Okay, I admit that I lack some experience in this area of mathematics.
(z is the surface? I thought it was the room coordinate?)

Maybe I'd figure it out eventually, but not today.

@ Mini-Geek: thanks!