Let $a(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).

Let $T(n, k)$ be A083906. Here

$$ T(n, k) = [q^k]\sum\limits_{m=0}^{n} \binom{n}{m}_q $$

where $\binom{n}{m}_q$ denotes Gaussian binomial coefficient.

I conjecture that

$$ T(n, k) = [n > k](T(n-1, k) + a(k)) + [n \leqslant k](2T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1)), \\ T(0, k) = [k = 0], T(1, k) = 2[k = 0]. $$

where square bracket denotes Iverson bracket.

I also conjecture that

$$ \sum\limits_{i = 0}^{n}T(n-i, i) = a(n+1) $$

Note that upper limit of the summation can be reduced.

Is there a way to prove it? Is there a way to get a simple closed form for $T(n,k)$ based on the given recursion?