Understanding MFCC

I am not from electrical or electronic background so my knowledge will be lacking.

MFCC is represented by 39 values for each window frame. 12 values are the mel filter-bank and we get 13th value by taking DCT[ Is this right ]?
So rest are the delta and double delta and their energy.

Below is the equation for calculating mel frequency cepstrum:

It appears to me that it gives a single value for a window frame. I understand that S[m] is the log energies for each M filter. Does c[n] refers to coefficients of n the frame?

Isn’t the equation 6.145 summing the log energies over M filters?. If there are 13 mel filters(M=13) then equation 6.145 appears to be the sum of 13 log energies which gives 1 value. Isn’t this logic right.

I need to understand how 13 values are found from equation 6.145