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I use data like this

data = [253993,275396.2,315229.5,356949.6,400158.2,442431.7,495102.9,570164.8,\
640993.1,704250.4,767455.4,781807.8,776332.3,794161.7,834177.7,931651.5,\
1028390,1114914]
Then, I import statsmodels and use Holt’s Method 
import matplotlib.pyplot as plt
from statsmodels.tsa.holtwinters import ExponentialSmoothing, SimpleExpSmoothing, Holt
# Holt’s Method
fit1 = Holt(data).fit(smoothing_level=0.8, smoothing_slope=0.2, optimized=False)
l1, = plt.plot(list(fit1.fittedvalues) + list(fit1.forecast(5)), marker='o')
fit2 = Holt(data, exponential=True).fit(smoothing_level=0.8, smoothing_slope=0.2, optimized=False)
l2, = plt.plot(list(fit2.fittedvalues) + list(fit2.forecast(5)), marker='o')
fit3 = Holt(data, damped=True).fit(smoothing_level=0.8, smoothing_slope=0.2)
l3, = plt.plot(list(fit3.fittedvalues) + list(fit3.forecast(5)), marker='o')
l4, = plt.plot(data, marker='o')
plt.legend(handles = [l1, l2, l3, l4], labels = ["Holt's linear trend", "Exponential trend", "Additive damped trend", 'data'], loc = 'best', prop={'size': 7})
plt.show()
An exception was thrown in fit2
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-17-9ce7957db4db> in <module>()
      3 l1, = plt.plot(list(fit1.fittedvalues) + list(fit1.forecast(5)), marker='o')
----> 5 fit2 = Holt(data, exponential=True)
      6 fit2.fit(smoothing_level=0.8, smoothing_slope=0.2, optimized=False)
      7 l2, = plt.plot(list(fit2.fittedvalues) + list(fit2.forecast(5)), marker='o')
g:\competition\venv\lib\site-packages\statsmodels\tsa\holtwinters.py in __init__(self, endog, exponential, damped)
    851     def __init__(self, endog, exponential=False, damped=False):
    852         trend = 'mul' if exponential else 'add'
--> 853         super(Holt, self).__init__(endog, trend=trend, damped=damped)
    855     def fit(self, smoothing_level=None, smoothing_slope=None, damping_slope=None, optimized=True):
g:\competition\venv\lib\site-packages\statsmodels\tsa\holtwinters.py in __init__(self, endog, trend, damped, seasonal, seasonal_periods, dates, freq, missing)
    389         self.trending = trend in ['mul', 'add']
    390         self.seasoning = seasonal in ['mul', 'add']
--> 391         if (self.trend == 'mul' or self.seasonal == 'mul') and (endog <= 0.0).any():
    392             raise NotImplementedError(
    393                 'Unable to correct for negative or zero values')
TypeError: '<=' not supported between instances of 'list' and 'float'
I don't know why, others are normal.(Holt-Winters’ Method like this too)
I think it's the exponential parameter that's causing the problem.So what should I do to use an exponential model?
I'm not very experienced with this library but it appears to want a series rather than a list for data.  Bring in pandas.pd and convert your data to a pd.Series:
import matplotlib.pyplot as plt
from statsmodels.tsa.holtwinters import ExponentialSmoothing, SimpleExpSmoothing, Holt
import pandas as pd
data = [253993,275396.2,315229.5,356949.6,400158.2,442431.7,495102.9,570164.8,\
640993.1,704250.4,767455.4,781807.8,776332.3,794161.7,834177.7,931651.5,\
1028390,1114914]
series = pd.Series(data)
# Holt's Method
fit1 = Holt(series).fit(smoothing_level=0.8, smoothing_slope=0.2, optimized=False)
l1, = plt.plot(list(fit1.fittedvalues) + list(fit1.forecast(5)), marker='o')
fit2 = Holt(series, exponential=True).fit(smoothing_level=0.8, smoothing_slope=0.2, optimized=False)
l2, = plt.plot(list(fit2.fittedvalues) + list(fit2.forecast(5)), marker='o')
fit3 = Holt(series, damped=True).fit(smoothing_level=0.8, smoothing_slope=0.2)
l3, = plt.plot(list(fit3.fittedvalues) + list(fit3.forecast(5)), marker='o')
l4, = plt.plot(series, marker='o')
plt.legend(handles = [l1, l2, l3, l4], labels = ["Holt's linear trend", "Exponential trend", "Additive damped trend", 'data'], loc = 'best', prop={'size': 7})
plt.show()
        
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