Sum of Square,Square of Sum

Relevant wiki: Stirling's Formula

$\begin{aligned} B & = \lim_{n \to \infty} \frac {n^A \sum_{k=0}^n \binom nk^2}{\left(\sum_{k=0}^n \binom nk\right)^2} \\ & = \lim_{n \to \infty} \frac {n^A \binom {2n}n}{\left(2^n\right)^2} \\ & = \lim_{n \to \infty} \frac {n^A (2n)!}{2^{2n} (n!)^2} & \small \color{#3D99F6} \text{By Stirling's formula: }n! \sim \sqrt{2\pi n}\left(\frac ne\right)^n \\ & = \lim_{n \to \infty} \frac {n^A \sqrt{4\pi n}\left(\frac {2n}e\right)^{2n}}{2^{2n} \left(\sqrt{2\pi n}\left(\frac ne\right)^n\right)^2} \\ & = \lim_{n \to \infty} \frac {2^{2n+1}\pi^\frac 12 n^{A+\frac 12 + 2n}e^{2n}}{2^{1+2n}\pi n^{1+ 2n}e^{2n}} \\ & = \lim_{n \to \infty} \frac {n^{A+\frac 12}}{\sqrt \pi n} & \small \color{#3D99F6} \text{For }B \text{ to exist and }\ne 0, A = \frac 12 \\ & = \frac 1{\sqrt \pi} \end{aligned}$

Therefore, $A+B = \dfrac 12 + \dfrac 1{\sqrt \pi} \approx \boxed{1.064}$ .