The total number of ways, including triplets, is 15!/(3!)^5 = 168,168,000

The Python itertools.permutations does not account for duplicate elements, but a different library does have that function:    more-itertools.distinct_permutations.

A test run counted the expected number.

But when all triplets are excluded, the count is 145,895,280

The first 10 and the last 10 permutations are shown:

['AABABBCCDCDEDEE',

 'AABABBCCDCDEEDE',

 'AABABBCCDCEDDEE',

 'AABABBCCDCEDEDE',

 'AABABBCCDCEDEED',

 'AABABBCCDCEEDDE',

 'AABABBCCDCEEDED',

 'AABABBCCDDCEDEE',

 'AABABBCCDDCEEDE',

 'AABABBCCDDECDEE',

 ... ... ... ...  ,

 ... ... ... ...  ,

 ... ... ... ...  ,

 'EEDEDDCCBBACBAA',

  'EEDEDDCCBBCAABA',

  'EEDEDDCCBBCABAA',

  'EEDEDDCCBCAABAB',

  'EEDEDDCCBCAABBA',

  'EEDEDDCCBCABAAB',

  'EEDEDDCCBCABABA',

  'EEDEDDCCBCABBAA',

  'EEDEDDCCBCBAABA',

  'EEDEDDCCBCBABAA']]

------------------------

count = 0

results = []

from more_itertools import distinct_permutations

for p in  (distinct_permutations('AAABBBCCCDDDEEE')):

    skip = False

    inarow = 0

    for i,v in enumerate(p):

        if i == 0:

            last = v

            inarow = 1

            continue

        if v == last:

            inarow += 1

            if inarow >= 3:

                skip = True

                break

        elif v != last:

            inarow = 1

            last = v

    if not skip:

        results.append(''.join(p))

        count += 1

print(count)