Beta Function

B (α, β) = {[\frac{Γ (α + β)}{Γ (α) Γ (β)}]}^{- 1}

Inverse to denote the relationship with combinatorics, Gamma function is the continuous factorial, so it is like the inverse of $(\binom{n}{k})$

If the gamma function was not weirdly shifted by one, it would look like

B (α, β) = \frac{Γ (α) Γ (β)}{Γ (α + β) (α + β + 1)}

which is one argument for the weird gamma function. Another one is the fact that $Γ (0) = - 1! = \infty$