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Not sure if it's the best way, but you can do the following:

sage: for n in range(10):
sage:    c = (x2^n*y2^n*v).coefficients()[0]
sage:    print('c_{} = {}'.format(n, c))
c_0 = 1
c_1 = -3
c_2 = 24
c_3 = -360
c_4 = 8640
c_5 = -302400
c_6 = 14515200
c_7 = -914457600
c_8 = 73156608000
c_9 = -7242504192000