

from __future__ import annotations





def 				__a							(       __A						):
    SCREAMING_SNAKE_CASE__         =    {1}
    SCREAMING_SNAKE_CASE__         =    [1]
    for _ in range(2			,							depth + 1						):
        SCREAMING_SNAKE_CASE__         =    [*previous_coefficients, 0]
        SCREAMING_SNAKE_CASE__         =    [0, *previous_coefficients]
        SCREAMING_SNAKE_CASE__         =    []
        for x, y in zip(__A			,							__A						):
            coefficients.add(x + y						)
            previous_coefficients.append(x + y						)
    return coefficients





def 				__a							(       __A						):
    SCREAMING_SNAKE_CASE__         =    set()
    for number in unique_coefficients:
        SCREAMING_SNAKE_CASE__         =    2
        SCREAMING_SNAKE_CASE__         =    number
        while divisor**2 <= copy_number:
            SCREAMING_SNAKE_CASE__         =    0
            while copy_number % divisor == 0:
                copy_number //= divisor
                multiplicity += 1
            if multiplicity >= 2:
                non_squarefrees.add(__A						)
                break
            divisor += 1

    return unique_coefficients.difference(__A						)





def 				__a							(       __A = 51						):
    SCREAMING_SNAKE_CASE__         =    get_pascal_triangle_unique_coefficients(__A						)
    SCREAMING_SNAKE_CASE__         =    get_squarefrees(__A						)
    return sum(__A						)


if __name__ == "__main__":
  print(f'''{solution() = }''')





from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve





def 				__a							(       __A			,							__A			,							__A						):
    SCREAMING_SNAKE_CASE__         =    np.stack((horizontal_flow, vertical_flow)			,							2						)

    # Create a grid of all pixel coordinates and subtract the flow to get the
    # target pixels coordinates
    SCREAMING_SNAKE_CASE__         =    np.stack(
        np.meshgrid(np.arange(0			,							image.shape[1]						)			,							np.arange(0			,							image.shape[0]						)						)			,							2						)
    SCREAMING_SNAKE_CASE__         =    np.round(grid - flow						).astype(np.intaa						)

    # Find the locations outside of the original image
    SCREAMING_SNAKE_CASE__         =    (grid < 0) | (grid >= np.array([image.shape[1], image.shape[0]]						))
    SCREAMING_SNAKE_CASE__         =    0

    SCREAMING_SNAKE_CASE__         =    image[grid[:, :, 1], grid[:, :, 0]]

    # Set pixels at invalid locations to 0
    SCREAMING_SNAKE_CASE__         =    0

    return warped





def 				__a							(       __A			,							__A			,							__A			,							__A = None			,							):
    if alpha is None:
        SCREAMING_SNAKE_CASE__         =    0.1

    # Initialize flow
    SCREAMING_SNAKE_CASE__         =    np.zeros_like(__A						)
    SCREAMING_SNAKE_CASE__         =    np.zeros_like(__A						)

    # Prepare kernels for the calculation of the derivatives and the average velocity
    SCREAMING_SNAKE_CASE__         =    np.array([[-1, 1], [-1, 1]]						) * 0.25
    SCREAMING_SNAKE_CASE__         =    np.array([[-1, -1], [1, 1]]						) * 0.25
    SCREAMING_SNAKE_CASE__         =    np.array([[1, 1], [1, 1]]						) * 0.25
    SCREAMING_SNAKE_CASE__         =    np.array(
        [[1 / 12, 1 / 6, 1 / 12], [1 / 6, 0, 1 / 6], [1 / 12, 1 / 6, 1 / 12]]						)

    # Iteratively refine the flow
    for _ in range(__A						):
        SCREAMING_SNAKE_CASE__         =    warp(__A			,							__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						) + convolve(__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						) + convolve(__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						) + convolve(__A			,							-kernel_t						)

        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						)

        # This updates the flow as proposed in the paper (Step 12)
        SCREAMING_SNAKE_CASE__         =    (
            derivative_x * avg_horizontal_velocity
            + derivative_y * avg_vertical_velocity
            + derivative_t
        )
        SCREAMING_SNAKE_CASE__         =    update / (alpha**2 + derivative_x**2 + derivative_y**2)

        SCREAMING_SNAKE_CASE__         =    avg_horizontal_velocity - derivative_x * update
        SCREAMING_SNAKE_CASE__         =    avg_vertical_velocity - derivative_y * update

    return horizontal_flow, vertical_flow


if __name__ == "__main__":
  import doctest

  doctest.testmod()


