Some restricted weighted sums I am interested in knowing the order of magnitude of the following two weighted sums. The first one is as follows:
Suppose $(w_1, w_2, \cdots, w_{n-1})$ are positive numbers, and suppose that $\lambda$ is a positive real number. Let $d$ be a given positive integer (with $d < \lfloor \lambda \rfloor$). Then I want to know the value of
$$\displaystyle \sum_{\substack{w_1 x_1 + \cdots + w_{n-1} x_{n-1} \leq \lambda, \text{  } x_1 < d}} 1 $$
where the sum is over non-negative integers $x_1, \cdots x_{n-1}$.
If $w_1 = w_2 = \cdots = w_{n-1} = 1$ and $\lambda$ is a positive integer, then the above sum can be expressed as $$\displaystyle \sum_{x_1 + \cdots + x_n = \lambda} 1 - \sum_{x_2 + \cdots + x_n = \lambda - d} 1,$$
which can be expressed in closed form:
$$\displaystyle \binom{\lambda + n - 1}{n-1} - \binom{\lambda - d + n - 1}{n-1}$$
which is equal to
$$\displaystyle \frac{(\lambda + n-1) \cdots (\lambda + 1)}{(n-1)!} - \frac{(\lambda - d + n-1) \cdots (\lambda - d + 1)}{(n-1)!} = \frac{d\lambda^{n-2}}{(n-2)!} + O(\lambda^{n-3}).$$
Now my question is for arbitrary weights $(w_1, \cdots, w_{n-1})$ and $\lambda$ an arbitrary positive integer, can the same estimate hold? In particular, I am asking if the following equality holds:
$$\displaystyle \sum_{\substack{w_1 x_1 + \cdots + w_{n-1} x_{n-1} \leq \lambda, \text{  } x_1 < d}} 1 = \frac{d\lambda^{n-2}}{w_1 \cdots w_{n-1} (n-2)!} + O(\lambda^{n-3})$$
A related question is whether the following weighted sum has a nice asymptotic form as well:
$$\displaystyle \sum_{\substack{w_1 x_1 + \cdots + w_{n-1} x_{n-1} \leq \lambda, \text{  } x_1 < d}} w_1 x_1 + \cdots + w_{n-1} x_{n-1} $$
I suspect, based on the answer provided at Values of various weighted sums , that the asymptotic should be
$$\displaystyle \frac{ d\lambda^{n-1}}{w_1 \cdots w_{n-1} (n-2)!} + O(\lambda^{n-2}).$$
A third sum I want to estimate is when the weight of the summands need not be the same as the original weight. In particular, for integral weights $w_1, \cdots, w_{n-1}$ and $v_1, \cdots, v_{n-1}$ positive real numbers not identical to $w_1, \cdots, w_{n-1}$ and $\lambda$ a positive integer:
$$\displaystyle \sum_{w_1 x_1 + \cdots + w_{n-1}x_{n-1} = \lambda, x_1 < d} v_1 x_1 + \cdots + v_{n-1} x_{n-1}$$
Here I suspect the answer to be
$$\displaystyle \frac{d\lambda^{n-2}}{w_1 \cdots w_{n-1}(n-2)!} \left(\frac{v_1}{w_1} + \cdots + \frac{v_{n-1}}{w_{n-1}}\right) + O(\lambda^{n-3})$$
Any help would be much appreciated.
 A: We might as well assume that $\lambda$ is an integer. Then your first
sum is the coefficient of $x^\lambda$ in
  $$ F(x) =\frac{1+x^{w_1}+x^{2w_1}+\cdots+x^{(d-1)w_1}}
      {(1-x^{w_2})\cdots(1-x^{w_{n-1}})(1-x)}. $$
The Laurent expansion of $F(x)$ at $x=1$ begins
   $$ F(x) = \frac{d}{w_2w_3\cdots w_{n-1}}\frac{1}{(1-x)^{n-1}}
       +\cdots, $$
so the coefficient of $x^\lambda$ is asymptotic to
   $$ \frac{d}{w_2\cdots w_{n-1}}{-(n-1)\choose \lambda} 
    \sim \frac{d}{w_2\cdots w_{n-1}}\frac{\lambda^{n-2}}
      {(n-2)!}, $$
agreeing with your first conjecture except that there is no $w_1$ in the
denominator.
Addendum. For the second sum, define
  $$ G(x,y) = \frac{1+x^{v_1}y^{w_1}+\cdots+x^{(d-1)v_1}y^{(d-1)w_1}}
     {(1-x^{v_2}y^{w_2})\cdots(1-x^{v_{n-1}}y^{w_{n-1}})}. $$
Then the second sum is equal to the coefficient of $y^\lambda$ in
  $$ \frac{d}{dx} G(x,y)|_{x=1} =
     \frac{v_1y^{w_1}+\cdots+(d-1)v_1y^{(d-1)w_1}}
      {(1-y^{w_2})\cdots (1-y^{w_{n-1}})} $$
  $$ + \sum_{k=2}^{n-1} \frac{(1+y^{w_1}+\cdots+y^{(d-1)w_1})v_k}
     {(1-y^{w_k})\cdot (1-y^{w_2})(1-y^{w_3})\cdots
     (1-y^{w_{n-1}})}. $$
The first term grows like $(1-y)^{-(n-2)}$ and thus is asymptotically
negligible compared to the remaining terms (the sum from $k=2$ to
$n-1$). Thus 
  $$ \frac{d}{dx} G(1,y) \sim d\sum_{k=2}^{n-1}
    \frac{v_k}{w_k(w_2 w_3\cdots w_{n-1})} \frac{1}{(1-y)^{n-1}}. $$
The coefficient of $y^\lambda$ is asymptotic to 
  $$ \frac{d}{w_2w_3\cdots w_{n-1}}\sum_{k=2}^{n-1}\frac{v_k}{w_k}
      \cdot \frac{\lambda^{n-2}}{(n-2)!}. $$
This agrees with the conjecture (up to computational error) except 
for the $w_1$ factor and $v_1/w_1$ term.
