Given the series expansion $$1+b_1\arctan x+b_2 \arctan^2 x=1+\sum_{k=1}^\infty a_k x^k$$ one has the relationships $$a_{2k+1}=(-1)^k\frac{b_1}{2k+1},$$ $$a_{2k}=(-1)^k b_2\sum_{i=0}^k \frac{1}{(2i+1)(2k-2i+1)}.$$
Carlo Beenakker
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