I would like to find a function f such that $$y_i=f(i) \geq 0$$ such that:
$$\sum_{i=0}^n y_i = C(1-\frac{3p}{4})^{n(1-\frac{3p}{4})}(\frac{3p}{4})^{n(\frac{3p}{4})},$$
where $$C > 0, 0 \leq p \leq 1,$$ and
$$\sum_{w=0}^n\sum_{k=0}^n\sum_{i=0}^n (-\frac{1}{3})^k\binom{n-i}{w-k}\binom{i}{k}3^{i}y_i$$ converges to a constant as $$n\rightarrow \infty.$$
Do you have any ideas? First of all, suggestions how to decompose the first sum would be useful.
Thanks.