We need to find out
$$\sum {\binom{N}{a_1,a_2,a_3...a_B} a_1^{\alpha}a_2^{\alpha}...a_C^{\alpha} }$$
$$a_1+a_2...a_B=N, \alpha>0 ,0\lt C \le B$$
All are nonnegative integers.
We need to sum for all $a_i$. $N$,$B$ and $C$ are constants.
I was wondering if a closed form or a recurrence exists. I tried to solve it and failed miserably. All suggestions are welcome.
On
The sum:
$$ \sum_{a_1 + a_2 + \dotsb + a_B = N} \binom{N}{a_1, a_2, \dotsc, a_B} x_1^{a_1} x_2^{a_2} \dotsm x_B^{a_B} = (x_1 + x_2 + \dotsm + x_B)^N $$
Note that if $A(z) = \sum_{n \ge 0} a_n z^n$, then:
$$ z \frac{\mathrm{d}}{\mathrm{d} z} A(z) = \sum_{n \ge 0} n a_n z^n $$
Doing this operation $\alpha$ times for each $x_i$, and setting $x_1 = x_2 = \dotsb = x_B = 1$ in the result gives your sum Ain't pretty...
The summation is difficult while finding all permutation of $a's$. For a similiar problem I found the Equation-15 useful in following paper:
https://cs.uwaterloo.ca/journals/JIS/VOL16/Eger/eger6.pdf
The sum is converted to simple incremental variable and product of binomial coefficients. Hope it would help you too.