Boxes and Objects!

Let the boxes be named A,B,C and D. Now, form a table to see how many objects can be taken by each box.

Now, it is a simple matter of counting each case and adding them all up.

The first and the third row have $\displaystyle \binom{6}{1}$ ways each.
The second row has $\displaystyle \binom{6}{3}$ ways.
Rows 4 to 7 have $\displaystyle \binom{6}{2}\cdot\binom{4}{1}$ ways each.
Rows 8 and 9 have $\displaystyle \binom{6}{4}\cdot\binom{2}{1}$ ways each.
And the last row has $\displaystyle \binom{6}{2}\cdot\binom{4}{2}\cdot\binom{2}{1}$ ways.

Adding all these answers we get $\displaystyle a_6=512$

The exponential generating function for ( $a_r$ ) is [ $\frac{e^x +e^{-x}}{2}$ ] [ $\frac{e^x +e^{-x}}{2}$ ] [ $\frac{e^x -e^{-x}}{2}$ ] [ $e^x$ ]

$\frac{1}{8}$ [ $e^{2x} - e^{-2x}$ ][ $e^{2x} +1$ ]

Which on summation from $r= 0$ to $\infty$ gives generalized value as

$a_r$ = $\frac{1}{8}$ { $4^r +2^r -(-2)^r$ }

Putting $r=6$ we get $a_6$ as $512$