Special Partitions 2

An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers. Partitions that differ only by the order of their summands are considered the same.
We call an integer partition special if 1) all its summands are distinct, and 2) all its even summands are also divisible by $4$.For example, the special partitions of $10$ are: \[10 = 1+4+5=3+7=1+9\]
The number $10$ admits many more integer partitions (a total of $42$), but only those three are special.
Let be $P(n)$ the number of special integer partitions of $n$. You are given that $P(1) = 1$, $P(2) = 0$, $P(3) = 1$, $P(6) = 1$, $P(10)=3$, $P(100) = 37076$ and $P(1000)=3699177285485660336$.
Find $\displaystyle \sum_{i=1}^{10^7} P(i)$. Give the result modulo $10^9+7$.

This question seems to be a competitive programming problem rather than a standard math problem. It involves knowledge of number theory and dynamic programming, a common technique in competitive programming. The problem can be understood as follows:

We need to calculate the number of special integer partitions for each number from $1$ to $10^7$ and sum them up. However, direct calculation is impossible due to the problem’s size. To simplify the problem, we need to consider the properties of special integer partitions:

1) All even summands (parts that add up to the number) are divisible by 4, thus, the even summands in the partition are $4, 8, 12, 16, \dots$.

2) All odd summands, irrespective of being divisible by 4 or any other number, should be unique. Hence, the odd summands are $1, 3, 5, 7, \dots$.

This gives an insight that we can apply dynamic programming to calculate the special partitions within the given range.

Let `DP[i][j]` be the number of special partitions of number `i`, using numbers no larger than `j`.

Base case, `DP[0][0] = 1`, represents one possible way to partition `0` using the maximum summand as `0`.

For each `i` (from 1 to $10^7$) and each `j` (next possible summand), there are two situations to be considered:

– If `j` is larger than `i`, then we cannot use `j` to partition the `i`, so `DP[i][j] = DP[i][j-1]`.

– If `j` is less or equal to `i`, we have two options. Either use `j` or not. Thus, `DP[i][j] = DP[i][j-1](not using j) + DP[i-j][j-2](use j)`.

Note that decrementing by `2` in the case where we use `j` is based on the requirement that all partitions must be distinct.

Next, calculate `DP[i][j]` for the entire table and take the sum as `Sigma`.

Lastly, to find the answer to the problem. ¿We calculate `Sigma % (10^9+7)`.

One thing to note: doing the calculation directly would require immense computational resources, an optimized code using a programming language like Python, C++, or Java would be needed. Also, using smaller problem size first is a good way of testing the code as verifying the result with $10^7$ size table would be practically impossible.

Remember that this answer does not provide a direct numerical solution, but is a prescription of how a solution can be obtained programmatically due to the scale of the problem.

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