Deriving the Convolution Kernel of the Inverse of a Signal
Since Convolution is a Linear Operation the easiest way to derive the coefficients is using a Matrix Representation of the problem.

Take the kernel $ y $ and build from it the Convolution Matrix.
Then write the vector representation of the Delta Funciton $ d $ and solve:

$$ Y x = d $$

In case the problem is solvable (Namely there is an inverse operator of $ Y $) you'll get you answer.
Of course you can always use Least Squares Solution.

Full code for building Convolution Matrix was given in Estimating a Signal Given a Noisy Measurement of the Signal and Its Derivative (Denoising) and Deconvolution of Synthetic 1D Signals - How To?