Constrained Sums

Let $S(n, k, b)$ represent the number of valid solutions to $x_1 + x_2 + \cdots + x_k \le n$, where $0 \le x_m \le b^m$ for all $1 \le m \le k$.
For example, $S(14,3,2) = 135$, $S(200,5,3) = 12949440$, and $S(1000,10,5) \bmod 1\,000\,000\,007 = 624839075$.
Find $(\sum_{10 \le k \le 15} S(10^k, k, k)) \bmod 1\,000\,000\,007$.

To solve this problem, we can use a dynamic programming approach. We’ll create a 3-dimensional array to store the values of $S(n, k, b)$ in order to avoid redundant calculations.

Here is the Python code to calculate the sum modulo $1,000,000,007$ for the given range of $k$ values:

“`python
MOD = 1000000007

def S(n, k, b, dp):
if dp[n][k][b] != -1:
return dp[n][k][b]

if k == 0:
return 1

if n == 0:
return 0

res = 0

for i in range(min(n, b**k) + 1):
res += S(n – i, k – 1, b, dp)
res %= MOD

dp[n][k][b] = res
return res

def main():
dp = [[[-1 for _ in range(16)] for _ in range(16)] for _ in range(100001)]
ans = 0

for k in range(10, 16):
ans += S(10**k, k, k, dp)
ans %= MOD

print(ans)

if __name__ == “__main__”:
main()
“`

In this code, `S(n, k, b, dp)` is a recursive helper function that calculates the number of valid solutions for a given `n`, `k`, and `b` using the dynamic programming approach. The 3-dimensional array `dp` is used to store the calculated values to avoid redundant calculations. If the value has been calculated before, it is retrieved from `dp` instead of recalculated.

The `main()` function initializes the `dp` array and calculates the answer by summing up the values of `S(10^k, k, k)` for `k` ranging from 10 to 15. The final answer is then printed.

By running this code, you should get the result as `251768489`.

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