  In this paper we explain the relation between the free energy of the SYK
model for $N$ Majorana fermions with a random $q$-body interaction and the
moments of its spectral density. The high temperature expansion of the free
energy gives the cumulants of the spectral density. Using that the cumulants
are extensive we find the $p$ dependence of the $1/N^2$ correction of the
$2p$-th moments obtained in 1801.02696. Conversely, the $1/N^2$ corrections to
the moments give the correction (even $q$) to the $\beta^6$ coefficient of the
high temperature expansion of the free energy for arbitrary $q$. Our result
agrees with the $1/q^3$ correction obtained by Tarnopolsky using a mean field
expansion. These considerations also lead to a more powerful method for solving
the moment problem and intersection-graph enumeration problems. We take
advantage of this and push the moment calculation to $1/N^3$ order and find
surprisingly simple enumeration identities for intersection graphs. The $1/N^3$
corrections to the moments, give corrections to the $\beta^8$ coefficient (for
even $q$) of the high temperature expansion of the free energy which have not
been calculated before. Results for odd $q$, where the SYK `Hamiltonian' is the
supercharge of a supersymmetric theory are discussed as well.
