Derivation of binomial coefficient from second order difference equation

Question

I am looking at this paper and I am trying to derive Eq. 1 from Eq. 2. It is said that for second order difference equation of the form, $$N(t+1)-N(t)=\alpha\Big[C_2(N(t))-C_2(N(t-1))\Big]$$ the solution is, $$C_2(N)=\frac{N(N-1)}{2}$$ which is equivalent of saying, $$\frac{N(N-1)}{2}\equiv\binom{N}{2}$$ for $N\geqslant2$. I wonder how to derive the solution $C_2(N)$ from the second order difference equation? Is this something known in literature that one can derive binomial coefficients from difference equations?

Answer 1

You're misreading the paper. It doesn't derive the binomial coefficients from the difference equation. It introduces the binomial coefficients in equation $(1)$ as the number of pairs, and then defines a “binary recombinant expansion process” as one which involves these binomial coefficients in a certain way as in equation $(2)$.