Interesting Combinatorics Problem! :D

listfor(n, 0, 15, ((2*n + 1)*(15 nCr n))==(1, 45, 525, 3185, 12285, 33033, 65065, 96525, 109395, 95095, 63063, 31395, 11375, 2835, 435, 31)>>>Sum==524,288

Interesting Problem!!!!!! 

$\text{ Let } S=$$15C0+3(15C1)+5(15C2)+7(15C3)+...+31(15C15)$

Then,  $S=(15C15)+3(15C14)+...+31(15C0)$     (Using the property that: $nCk = nC(n-k)$ ).

Adding the expressions above, to get:

$2S=(15C0)(31+1)+(15C1)(3+29)+...+(15C15)(31+1)$

Hence, $2S=32(15C0+15C1+15C2+...+15C15)=32*2^{15}=2^5*2^{15}=2^{20}$

Therefore, $S=2^{19}$