All possible tuples satisfying conditions

Let $\beta = (\beta_1, \beta_2, \ldots)$ be any given sequence of non-negative integers with all but finitely many $\beta_i$ zero. I want to collect all possible tuples $\beta^{\prime}= (\beta_1^{\prime}, \beta_2^{\prime}, \ldots)$ that satisfies the following two conditions:

(1) $\beta^{\prime} \geq \beta$, that is $\beta^{\prime}_i \geq \beta_i$ for all $i$

(2) $\beta^{\prime}$ can have more entries than $\beta$, in that case $\beta^{\prime}_i$ satisfies the condition $\sum i\cdot\beta^{\prime}_i = d$, where $n$ is a positive integer (for instance, if $\beta = (1), \text{and}~ d = 3$ then all possible $\beta^{\prime}$ will be the following list of tuples $\{ (1), (2), (3), (1,1) \}$).

Any suggestions on how to implement this in Mathematica would be great.

So, I tried to UseModule[{n}, n== Length[b], Table[ IF[conditions, betaprime ] range ]but it only gives me a nested lists out of which I couldn't able to list all possible $\beta^{\prime}$s.