I would like to determine the leading order in $n$ of this combination of binomial probabilities:
$$ f(n) = \sum_{k=0}^{n-1} \sum_{l=0}^{k} \sum_{m=0}^{k} \frac{P_l P_m}{P_k}$$
where $P_k = { n \choose k} p^k (1-p)^{n-k}$ with $p \in (0,1)$.
I checked it numerically and it seems that $ f(n) =1 + o(n)$.
I also tried converting the factorials into their Stirling approximations and the sums into integrals, but didn't get the finite limit.
Edit: numerical check was wrong... it's exponential.
Either you forgot one term or I am totally wrong.
$$F(n,p) = \sum_{k=0}^{n-1} \sum_{l=0}^{k} \sum_{m=0}^{k} \frac{P_l \,P_m}{P_k}\quad \quad\quad\text{where}\quad P_k = { n \choose k} p^k (1-p)^{n-k}$$
Using the Gaussian hypergeometric function $$F(n,p)=\sum_{k=0}^{n-1} \frac{\Big[ (1-p)^{k-n+1}-p^{k+1} \binom{n}{k+1} \, _2F_1\left(1,k-n+1;k+2;\frac{p}{p-1}\right)\Big]^2}{ \binom{n}{k} \,p^k\,(1-p)^{k-n+2}}$$
The table below reproduces the logarithms for various $n$ and $p$
$$\left( \begin{array}{cccccc} n & \log \left(F\left(n,\frac{1}{6}\right)\right)& \log \left(F\left(n,\frac{1}{3}\right)\right)& \log \left(F\left(n,\frac{1}{2}\right)\right)& \log \left(F\left(n,\frac{2}{3}\right)\right)& \log \left(F\left(n,\frac{5}{6}\right)\right) \\ 10 & 14.0527 & 8.12235 & 4.92756 & 2.91542 & 1.26060 \\ 20 & 31.2516 & 18.3394 & 10.9887 & 6.09719 & 2.79578 \\ 30 & 48.7561 & 28.8999 & 17.4681 & 9.62378 & 4.16702 \\ 40 & 66.3824 & 39.5888 & 24.0913 & 13.3396 & 5.58680 \\ 50 & 84.0747 & 50.3463 & 30.7880 & 17.1443 & 7.09347 \\ 60 & 101.809 & 61.1465 & 37.5296 & 21.0000 & 8.67212 \\ 70 & 119.571 & 71.9759 & 44.3017 & 24.8891 & 10.3005 \\ 80 & 137.354 & 82.8266 & 51.0958 & 28.8019 & 11.9625 \\ 90 & 155.154 & 93.6935 & 57.9064 & 32.7324 & 13.6479 \\ 100 & 172.965 & 104.573 & 64.7302 & 36.6767 & 15.3506 \\ 200 & 351.446 & 213.736 & 133.341 & 76.5084 & 32.8285 \\ 300 & 530.216 & 323.190 & 202.247 & 116.642 & 50.6368 \\ 400 & 709.104 & 432.763 & 271.272 & 156.898 & 68.5723 \\ 500 & 888.056 & 542.400 & 340.363 & 197.219 & 86.5761 \\ \end{array} \right)$$ which are almost linear functions of $n$.