df['bin'] = pd.cut(df.age, [0,4,9,14])
pvtdf = df.pivot_table(index='type', columns=['bin'], values='days',
aggfunc=('count', 'sum')).fillna(0)
# count sum
# bin (0, 4] (4, 9] (9, 14] (0, 4] (4, 9] (9, 14]
# type
# a 2.0 0.0 1.0 6.0 0.0 1.0
# b 3.0 0.0 0.0 9.0 0.0 0.0
# c 0.0 1.0 0.0 0.0 1.0 0.0
# d 0.0 1.0 0.0 0.0 4.0 0.0
# e 0.0 0.0 1.0 0.0 0.0 2.0
# f 0.0 1.0 0.0 0.0 0.0 0.0