Palindromic coin flips

Discrete Mathematics Level pending

Given 11 flips of a fair coin, the probability of getting a palindromic sequence is \[\frac {a}{b}\]. What is \[(a+b)^{2}\]?

Details and assumptions
1.A palindromic sequence is a series of coin flips that are palindromic, such as TTTTT, HTHTHTH or HHTTHHTTTTHHTTHH.
2.The sequence does not matter (i.e TTTHTTT=TTTHTTT)


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