Let $1(n)$ be the number of digits $1$ in binary representation of number $n$.


For example, $13=1101\_2$ so $1(13)=3\\$


Is there explicit form of $\,\,\sum{1(i)x^i} $?


I checked OEIS and didn't find information about this sequence (1,1,2,2,2,3,1,2,2,3,...)



Number of 1 in binary representation of n
The generating function you ask about, more typically written as $\sum\_{i\ge 0} s\_2(i) x^i$, can be expressed as 
$$\frac{1}{1-x} \sum\_{m\ge 0} \frac{x^{2^m}}{1+x^{2^m}}$$ 
The number of 1's in the binary expansion is just the sums of digits; there also exists a generalization for sums of digits function in other bases. For details, references and further information, see for example ["Generating Functions for the Digital Sum and Other Digit Counting Sequences"](http://www.emis.de/journals/JIS/VOL12/Ruskey2/ruskey14.pdf) by Adams-Watters and Ruskey (Journal of Integer Sequences, 2009)


