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This article considers fully connected neural networks with Gaussian random weights and biases as well as $L$ hidden layers, each of width proportional to a large parameter $n$. For polynomially bounded non-linearities we give sharp estimates in powers of $1/n$ for the joint cumulants of the network output and its derivatives. Moreover, we show that network cumulants form a perturbatively solvable hierarchy in powers of $1/n$ in that $k$-th order cumulants in one layer have recursions that depend to leading order in $1/n$ only on $j$-th order cumulants at the previous layer with $j\leq k$. By solving a variety of such recursions, however, we find that the depth-to-width ratio $L/n$ plays the role of an effective network depth, controlling both the scale of fluctuations at individual neurons and the size of inter-neuron correlations. Thus, while the cumulant recursions we derive form a hierarchy in powers of $1/n$, contributions of order $1/n^k$ often grow like $L^k$ and are hence non-negligible at positive $L/n$. We use this to study a somewhat simplified version of the exploding and vanishing gradient problem, proving that this particular variant occurs if and only if $L/n$ is large. Several key ideas in this article were first developed at a physics level of rigor in a recent monograph of Daniel A. Roberts, Sho Yaida, and the author. This article not only makes these ideas mathematically precise but also significantly extends them, opening the way to obtaining corrections to all orders in $1/n$.