To come up with the emission per child, the original source Reproduction and the carbon legacies of individuals Global Environmental Change Volume 19, Issue 1, February 2009, pages 14-20, is considering an infinite series.
In other words, having one child results in all the lifetime emission of said child, plus all that child's descendants' emissions. Since each child usually has two biological parents, 4 grandparents, etc., to avoid double counting, the reference weights descendants' emissions by 0.5^n, where "n" is the number of generations into the future. For example n=1 for children, n=2 for grandchildren, etc.
This infinite series is potentially divergent, if fertility rates are above 2. So the reference assumes that the fertility rate will drop to 1.85 everywhere in the world by 2050, permitting the series to be convergent.