Eccoci.

A Slider is a chess piece that can move one square left or right.

This problem uses a cylindrical chess board where the left hand edge of the board is connected to the right hand edge. This means that a Slider that is on the left hand edge of the chess board can move to the right hand edge of the same row and vice versa.

Let $L(N,K)$ be the number of ways $K$ non-attacking Sliders can be placed on an $N \times N$ cylindrical chess-board.

For example, $L(2,2)=4$ and $L(6,12)=4204761$.

Find $L(10^9,10^{15}) \bmod \left(10^7+19\right)^2$.