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Constrained Sums

Published on Saturday, 3rd October 2015, 07:00 pm; Solved by 221;
Difficulty rating: 60%

Problem 528

Let S(n,k,b) represent the number of valid solutions to x1 + x2 + ... + xkn, where 0 ≤ xmbm for all 1 ≤ mk.

For example, S(14,3,2) = 135, S(200,5,3) = 12949440, and S(1000,10,5) mod 1 000 000 007 = 624839075.

Find (10 ≤ k ≤ 15 S(10k,k,k)) mod 1 000 000 007.