How to synthesize 1,3-pentadiene from butanal?
I believe one way would be to start with butanal participating in a Wittig reaction to form 1-pentene. 

$\ce{CH3CH2CH2CHO + H2C=PPh3 -&gt; CH3CH2CH2CH=CH2}$ 

Then allylic bromination, as you said. 

$\ce{CH3CH2CH2CH=CH2 -&gt;[{NBS}][{hv}] CH3CH2CHBrCH=CH2}$

Dehydrohalogenation with a hindered base like t-butoxide (to favor the elimination reaction) should give the diene. The only thing I'm not sure about is whether it would be E1 or E2. It isn't very hindered, but it is a secondary halide. With those factors, plus the alkoxide, I think it will be an E2 mechanism. 

Let me know if I made a mistake somewhere.