Evaluating $\frac{13}{1.2 .3 .2}+\frac{26}{2.3 .4 .4}+\frac{43}{3.4 .5 .8}+\frac{64}{4.5 .6 .16}+\cdots$ $$\frac{13}{1.2 .3 .2}+\frac{26}{2.3 .4 .4}+\frac{43}{3.4 .5 .8}+\frac{64}{4.5 .6 .16}+\cdots$$
I can reduce it to the general term,
$$\sum_{r=1}^\infty \frac{2r^2 + 7r +4}{r(r+1)(r+2)2^r}$$
I don't know how to go about this any further though. I also ran this in python and the sum is exceeding $1.5$ for $10,000$ terms, which is weird since it should converge to $1.5$, so it makes me doubt if the general term I've written is correct.

Consider $$f(x)=\sum_{r=1}^\infty \frac{2r^2 + 7r +4}{r(r+1)(r+2)}x^r=\sum_{r=1}^\infty \left(\frac{1}{r+1}-\frac{1}{r+2}+\frac{2}{r} \right)x^r$$
$$f(x)=\frac{\log (1-x)}{x^2}+\frac{1}{x}-\frac{\log (1-x)}{x}-2 \log
   (1-x)-\frac{1}{2}$$
$$f\left(\frac{1}{2}\right)=\frac 3 2$$
