The infinite number of darts are not very friendly. I suppose dynamic programming is not applicable for larger input. My DP approach got TLE #7, #8 and MemoryError from #9 onwards. If you're interested, here is part of my code snippet:

mod = int(1e9) + 9
darts = [1] + [0] * N
for n in xrange(N+1):
    for m in xrange(1, 61):
        if n + m > N:
            break
        darts[n+m] += darts[n] * one_dart[m]
        darts[n+m] %= mod

EDIT: Learn about linear recurrence relation matrix (of size ), and then think about constructing the matrix (of size ) that can be feeded to a cumulative vector. Start with the vector [1] + [0] * 60. Perform matrix exponentiation (divide and conquer, very straightforward) that is . And finally, pick the double ending elements from the final vector of . Problem 114 has a relevant approach.