Pandas is calculating the UNBIASED estimator of the excess Kurtosis. Kurtosis is the normalized 4th central moment. To find the unbiased estimators of the cumulants you need the k-statistics.

So the unbiased estimator of kurtosis is (k4/k2**2)

To illustrate this:

import pandas as pd
import numpy as np

np.random.seed(11234)
test_series = pd.Series(np.random.randn(5000))
test_series.kurtosis()
#-0.0411811269445872

Now we can calculate this explicitly using the k-statistics:

n = len(test_series)
S1 = test_series.pow(1).sum()
S2 = test_series.pow(2).sum()
S3 = test_series.pow(3).sum()
S4 = test_series.pow(4).sum()

# Eq (7) and (5) from the k-statistics link
k4 = (-6*S1**4 + 12*n*S1**2*S2 - 3*n*(n-1)*S2**2 -4*n*(n+1)*S1*S3 + n**2*(n+1)*S4)/(n*(n-1)*(n-2)*(n-3))
k2 = (n*S2-S1**2)/(n*(n-1))
# k2 is the same as the N-1 variance: test_series.std(ddof=1)**2

k4/k2**2
#-0.04118112694458816

If you want better agreement to more decimal places, you'll need to be careful with the sums as they get rather large. But they're identical to 12 places.