http://endlesspint.com/2016-09-14-project-euler-versus-ep8/

lattice_less_naive.py

import numpy as np

def lat_path(x):
    
    grid = (x+1) * (x+1)

    test_m = np.zeros(grid).reshape(x+1, x+1)

    for row in range(x+1):
        for col in range(x+1):
            if row == 0 or col == 0:
                test_m[row][col] = 1
            else:
                test_m[row][col] = test_m[row-1][col] + test_m[row][col-1]
        
    print test_m
    print ""
    print "%d x %d grid has %d paths" % (x, x, test_m[x][x])

lat_path(5)

# [[   1.    1.    1.    1.    1.    1.]
# [   1.    2.    3.    4.    5.    6.]
# [   1.    3.    6.   10.   15.   21.]
# [   1.    4.   10.   20.   35.   56.]
# [   1.    5.   15.   35.   70.  126.]
# [   1.    6.   21.   56.  126.  252.]]

# 5 x 5 grid has 252 paths

lattice_naive.py

# change `boxes` variable to get path counts for lattice
# https://projecteuler.net/problem=15

boxes = 4
max_dist = boxes * 2

steps = {0: [(0,0)]}

for step in range(1, max_dist+1):
    steps[step] = []

#     print step-1
#     print steps[step-1]
#     print steps[step-1][-1]
    
    for path in steps[step-1]:
        if step-1==0:

            if steps[step-1][-1][0] < boxes:
                pre_step1 = list(steps[step-1])
                pre_step1.extend([(pre_step1[0][0]+1,pre_step1[0][1])])
                steps[step].append(pre_step1)

            if steps[step-1][-1][1] < boxes:
                pre_step2 = list(steps[step-1])
                pre_step2.extend([(pre_step2[0][0],pre_step2[0][1]+1)])
                steps[step].append(pre_step2)
        else:

            if path[-1][0] < boxes:
                pre_step1 = list(path)
                pre_step1.extend([(pre_step1[-1][0]+1,pre_step1[-1][1])])
                steps[step].append(pre_step1)

            if path[-1][1] < boxes:
                pre_step2 = list(path)
                pre_step2.extend([(pre_step2[-1][0],pre_step2[-1][1]+1)])
                steps[step].append(pre_step2)
    
        

print len(steps[max(steps.keys())])