Let $N$ be the total space of the vector bundle $\mathscr O_{\mathbb P^1}(-1)\oplus \mathscr O_{\mathbb P^1}(-1)$ over $\mathbb P^1$, and let $C_0\subset N$ be the zero section. Then $N$ is a quasi-projective Calabi-Yau threefold, and the moduli space of stable maps $\overline{\mathcal M}_g(N,[dC_0])$ is quasi-projective. It contains, as an open and closed substack, the space $$\overline{\mathcal M}_g(\mathbb P^1,d)\subset \overline{\mathcal M}_g(N,[dC_0]).$$ Faber and Pandharipande, in this paper on Hodge integrals, compute the contribution of the stable maps to $N$ factoring through the zero section $C_0\cong\mathbb P^1$ to the Gromov-Witten invariants of $N$. If the data $$\pi:U\to \overline{\mathcal M}_g(\mathbb P^1,d),\qquad f:U\to\mathbb P^1$$ describe the universal stable map to $\mathbb P^1$, this contribution is defined to be: $$C(g,d)=\int_{[\overline{\mathcal M}_g(\mathbb P^1,d)]^{\textrm{vir}}}e(R^1\pi_\ast f^\ast N).$$ In the paper, they write explicitely the generating series for these contributions in degree $1$. It is: $$\sum_{g\geq 0}C(g,1)t^{2g}=\Bigl(\frac{t/2}{\sin(t/2)} \Bigr)^2.$$

Question. Does anybody know (where to find) a similar formula for the generating series $\sum_{g\geq 0}C(g,d)t^{2g}$ when $d>1$? Maybe, it is hidden in that same paper, but I cannot see it. However, I am pretty sure this has been computed, as the Gromov-Witten theory of the local $\mathbb P^1$, as far as I know, is well understood.

Thank you very much!