the integrand should be

call the integral and substitute and for . then the Jacobian would be and thus

where , as usual, is the beta function. it's easy now to see that

Printable View

- Dec 24th 2010, 05:23 AMNonCommAlg
- Dec 24th 2010, 06:17 AMPaulRS
Consider the more general:

First note that what we are doing is choosing subsets of size by the following procedure:

Fix

**1.**Choose a subset of size .

**2.**Now choose a subset of size from the subset chosen in**1**- if possible.

It is clear that we are overcounting, but by how much? Well, any subset of size is part of subsets of size - that's the number of ways of completing the set -, that is

So in fact we have

Now plug and we get

**EDIT:**I am going to add a couple of problems for you to enjoy. (Rofl)

**1.**where is Euler's Totien function.

**2.**This one is beautiful too, let we define to be the number of pairs of integers that satisfy and - such a pair is called an inversion.

Show that:

What is the average number of inversions ?

See here - Dec 24th 2010, 10:20 AMDrexel28
- Dec 24th 2010, 01:48 PMBruno J.
- Dec 24th 2010, 03:26 PMDrexel28
- Dec 25th 2010, 04:16 AMPaulRS
- Dec 26th 2010, 03:27 AMUnbeatable0
Here are a few sums for you to calculate. However these may be too easy...

and one that involves much more algebra:

Where is the 'th harmonic number. - Dec 26th 2010, 12:34 PMchiph588@
- Dec 26th 2010, 03:03 PMchiph588@
Here's my rebuttal:

For , evaluate . - Dec 26th 2010, 03:19 PMDrexel28
- Dec 26th 2010, 04:30 PMchiph588@
- Dec 26th 2010, 04:34 PMDrexel28
- Dec 26th 2010, 06:40 PMBruno J.
- Dec 26th 2010, 07:05 PMDrexel28
- Dec 26th 2010, 08:05 PMchiph588@