Multilevel partial wavelet reconstruction with PyWavelets
I managed to write my own version of the wrcoef function which appears to work as expected:

import pywt
import pdb
import numpy as np

def wrcoef(X, coef_type, coeffs, wavename, level):
    N = np.array(X).size
    a, ds = coeffs[0], list(reversed(coeffs[1:]))

    if coef_type =='a':
        return pywt.upcoef('a', a, wavename, level=level)[:N]
    elif coef_type == 'd':
        return pywt.upcoef('d', ds[level-1], wavename, level=level)[:N]
    else:
        raise ValueError("Invalid coefficient type: {}".format(coef_type))



level = 4
X = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
coeffs = pywt.wavedec(X, 'db1', level=level)
A4 = wrcoef(X, 'a', coeffs, 'db1', level)
D4 = wrcoef(X, 'd', coeffs, 'db1', level)
D3 = wrcoef(X, 'd', coeffs, 'db1', 3)
D2 = wrcoef(X, 'd', coeffs, 'db1', 2)
D1 = wrcoef(X, 'd', coeffs, 'db1', 1)
print A4 + D4 + D3 + D2 + D1

# Results:
[  9.99200722e-16   1.00000000e+00   2.00000000e+00   3.00000000e+00
   4.00000000e+00   5.00000000e+00   6.00000000e+00   7.00000000e+00
   8.00000000e+00   9.00000000e+00   1.00000000e+01   1.10000000e+01
   1.20000000e+01   1.30000000e+01   1.40000000e+01   1.50000000e+01
   1.60000000e+01   1.70000000e+01]