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I've added another bit on:

sage: def c(k):
....: if k<3:
....: return 0
....: if k>2:
....: return sum_of_coeffs(nil(A,k))/(2*k)

where $c(k)$ returns the number of simple cycles in G of length $k$.

I've added another bit on:

def c(k): 
  ....:      if k<3: 
  ....:          return 0 
  ....:      if k>2: 
  ....:          return sum_of_coeffs(nil(A,k))/(2*k)  
sum_of_coeffs(nil(A,k))/(2*k)