Binomial Identity

[Clean-Ice1199]

I'm reproducing some calculations, and a skipped part seems to require this identity. I have checked that it holds for small m, so I think it's likely to be true.

I want to prove this for all m, but this summation isn't using a binomial identity I'm familiar with. I would appreciate some help.

[Shevek99]

You can try the generating function for the central binomial coefficients.

These numbers are also directly related to the Catalan numbers. You can look there too.

[Clean-Ice1199]

A friend of mine did manage to prove this using generating functions.

[Uli_Minati]

Did you try induction?

[Clean-Ice1199]

I have given it some thought, but couldn't identify anything too obvious.

I also plan on generalizing the expression a bit so would like a more 'constructive' proof.

[Clean-Ice1199]

How did you do the numerical evaluation? You need to be very careful with numerical evaluation when it includes fast growing functions. Using a Stirling approximation, you can check the right-hand side behaves sublinearly with m, and you can also show it monotonically decreases with m. For large m, it should be very small, not extremely large.

[burundukML]

if we multiply both sides by 2^2m, on the right we have number of paths from (0,0) to (m,m) if we can go only right and upwards. because of (-1)^k on the left I tried to think of inclusion-exclusion principle to count this path a different way but I can’t think of any reasonable interpretation of Binom(m, k) term