Maths problem

[thylacine]

[ QUOTE ]
Has anyone thought of a combinatorial solution to this ?
I have tried and failed .

We wish to prove :

2nCn + 2c1*(2n-2)C(n-1) + 4c2*(2n-4)C(n-2) + ...+ 2nCn = 4^n

[/ QUOTE ]

I don't know of one. I was thinking of the generating function method. But it would be interesting to know.

[blah_blah]

you may find the discussion here interesting

http://www.artofproblemsolving.com/F...00&t=40150

[thylacine]

[ QUOTE ]
you may find the discussion here interesting

http://www.artofproblemsolving.com/F...00&t=40150

[/ QUOTE ]

link didn't work

[jay_shark]

Here is a neat combinatorial argument .

Start from the origin and consider taking 2n steps on the lattice points of the line y=x or y=-x . Clearly the total number of paths is 2^(2n) . The number of ways of reaching the point (2n,0) is 2nCn . It is also true that the number of paths that do not intersect the x-axis is 2nCn . Since we can reflect all subsequent paths after the first step in the line y=1 or -1 to produce a bijection of paths that do not intersect the x-axis .

Consider the last path that intersects the x-axis at the points (2k,0) . The number of paths is 2kCk . Therefore we require the last (2n-2k) to be above or below the x-axis . The total number of ways is equal to the number of ways of reaching the point (2n,0) which is just (2n-2k)C(n-k) . Now sum over everything and we get our identity .

[jay_shark]

One important detail I left out .

We move in the direction up-right or down-right .

[TWCReborn]

To the original poster: Was this a homework problem in a combinatorics class? Just curious.

[thylacine]

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To the original poster: Was this a homework problem in a combinatorics class? Just curious.

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I made it up, but I'm sure this exact problem has been `made up' many times before. As above posters noted, it's closely related to generating function stuff for Catalan numbers.

[thylacine]

[ QUOTE ]
[ QUOTE ]
you may find the discussion here interesting

http://www.artofproblemsolving.com/F...00&t=40150

[/ QUOTE ]

link didn't work

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Nevermind, I tried it again and it worked this time (my crappy connection). Obviously this is an old maths problem as I suspected.

Hey blah_blah, I don't suppose that you are blahblahblah on that site you link?

[blah_blah]

[ QUOTE ]
Hey blah_blah, I don't suppose that you are blahblahblah on that site you link?

[/ QUOTE ]

one and the same