Taylor and Laplace

[where_was_i]

f(t)=e-t2. Find a taylor series for f(t) about x=0. Assuming it can be done term by term, find an expansion for Laplace{f}(s). If possible, sum the series. My series in the s-domain is 0!/(0!s)-2!/(1!s3)+4!/(2!s5)-6!/(3!s7)+...
I tried integrating and differentiating term by term to find a sum for it, but no luck. Is that series equal to a function? Can anyone verify that I did it right?