Let $T_k(x_1,\ldots,x_n)$ be the Todd polynomials, $e_k(x_1,\ldots,x_n)$ the elementary symmetric polynomials and $p_k(x_1,\ldots, x_n)$ the power sums of degree $k$.

We have the following generating formulas \begin{align*} \sum_{k\geq 0}T_k(x_1,\ldots,x_n)t^k = \prod_{i=1}^n\frac{tx_i}{1-e^{-tx^i}}\,,\\ \sum_{k\geq 0}e_k(x_1,\ldots,x_n)t^k = \prod_{i=1}^n(1+tx_i)\,,\\ \sum_{k\geq 0}\frac{1}{k!}p_k(x_1,\ldots,x_n)t^k = \sum_{i=1}^ne^{tx_i}\,. \end{align*} There is an explicit relation between $(1/k!)p_k$ and $e_k$ in terms of Newton's identities which can be expressed using generating series (see e.g. wikipedia). Is there some similar expression for $T_k$ in terms of $e_k$ or $p_k$?

For example if $X$ is a hyperkähler complex manifold. Replacing $x_i$ with $\alpha_i$ the roots of $c(TX)$, one can show that (see (3.13)): $$ td(X) = \text{exp}\Big[-2\sum_{n\geq0} b_{2n}\text{ch}_{2n}(TX)\Big]\,, $$ where $b_{2n}$ are the modified Bernoulli numbers. Is there an explicit formula without any assumptions on the geometry?