Background:
The underlying ring of the universal formal group law is, by a theorem of Quillen, the complex cobordism ring $\Omega_U^*$. Let $F_u(x,y) = x+y+\sum_{i,j\geq1}a_{ij}x^iy^j,\ a_{ij} \in \Omega_U^*$ be the universal group law and denote by FGL($R$) be the category of formal groups laws over the ring $R$. The universal formal group law $F_u$ can be seen as an element of FGL($\Omega_U^*\otimes \mathbb{Q}$) and by a structure theorem for the complex cobordism ring we have an isomorphism
$\Omega_U^* \otimes \mathbb{Q} \cong \mathbb{Q}[[\mathbb{P}^1],[\mathbb{P}^2],\dots]$
where $\mathbb{P}^i$ is the complex projective space.
Question:
Is there an explicit expression of the coefficients $a_{ij} \in \Omega_U^*\otimes\mathbb{Q}$ as polynomials in $\mathbb{Q}[[\mathbb{P}^1],[\mathbb{P}^2],\dots]$?
References:
http://www.map.mpim-bonn.mpg.de/Formal_group_laws_and_genera
http://www.map.mpim-bonn.mpg.de/Complex_bordism
http://people.virginia.edu/~mah7cd/Conferences/Vietnam2013/QuillenBP.pdf
By Mishenko's theorem the logarithm of the formal group law of complex cobordism over $\mathbb{Q}$ is $$ g_u(x)=x+\sum_{k=1}^\infty \frac{[\mathbb{C}\mathbb{P}^k]}{k+1}x^{k+1} $$ By definition of logarithm, $$ g_u(f_u(x,y))=g_u(x)+g_u(y), $$ i.e., $$ f_u(x,y)=g_u^{-1}(g_u(x)+g_u(y)). $$ In principle, the right hand side can be given an explict expression, e.g., by using Lagrange inversion theorem to write the Taylor coefficients of $g_u^{-1}(x)$, but I will not attempt determining a closed formula for the coefficient of $x^iy^j$ in $g_u^{-1}(g_u(x)+g_u(y))$.