Question

Consider the series:

2 6 20 70 252 …

now consider the matrix

1   1   1   1   1   1   …
1   2   3   4   5   6   …
1   3   6   10  15  21  …
1   4   10  20  35  56  …
1   5   15  35  70  126 …
1   6   21  56  126 252 …
…

• this is the matrix to find in how many number of ways you can reach (n,n) from (1,1) in an nXn matrix(where you can move only right or down)??

• the series is formed from the main diagonal elements of the matrix

how to generate the nth number in the series without matrix

Is there any generation function??

Answer 1

The diagonal is the Central binomial coefficients, with the closed equation: C(2*n,n) = (2*n)!/(n!)^2

More about it can be found here:
http://oeis.org/A000984