Sequences enumerating triangles of integer partitions

Define three conditions P, Q, and R on finite sequences of positive integers y as follows.
- y is of type P if y is an integer partition, i.e. y_i >= y_j for i < j.
- y is of type Q if y is a strict integer partition, i.e. y_i > y_j for i < j.
- y is of type R if y is a constant integer partition, i.e. y_i = y_j for i < j.

A triangle or twice-partition of type (C1,C2,C3) and weight n>=1 is a finite sequence of finite sequences of positive integers q=(q_1..q_k) satisfying:
- the reversed sorted concatenation of q is of type C1,
- the sequence of sums (Sum(q_1)..Sum(q_k)) is of type C2,
- each entry q_i is of type C3, and
- Sum_{i=1..k} Sum(q_i) = n.

Let A(C1,C2,C3)(n) be the number of triangles of type (C1,C2,C3) and weight n. We have the following 27 possibilities.

A(PPP) = A063834 =    1    3    6    15   28   66   122  266  503  1027 1913 3874 ...
A(PPQ) = A270995 =    1    2    4    7    12   23   37   64   108  180  290  488  ...
A(PPR) = A279784 =    1    3    5    12   18   40   60   121  186  344  524  955  ...
A(PQP) = A271619 =    1    2    5    8    18   34   65   109  223  386  698  1241 ...
A(PQQ) = A279785 =    1    1    3    4    7    11   18   28   47   71   108  166  ...
A(PQR) = A279786 =    1    2    4    5    9    16   22   28   49   69   94   138  ...
A(PRP) = A279787 =    1    3    4    10   8    29   16   64   58   124  57   469  ...
A(PRQ) = A279788 =    1    2    3    4    4    10   6    12   17   21   13   57   ...
A(PRR) = A279789 =    1    3    3    8    3    17   3    30   12   41   3    130  ...

A(QPP) = A279790 =    1    1    3    3    5    11   12   18   24   49   53   82   ...
A(QPQ) = A279790 =    1    1    3    3    5    11   12   18   24   49   53   82   ...
A(QPR) = A000009 =    1    1    2    2    3    4    5    6    8    10   12   15   ...
A(QQP) = A279375 =    1    1    3    3    5    9    12   16   24   39   49   70   ...
A(QQQ) = A279375 =    1    1    3    3    5    9    12   16   24   39   49   70   ...
A(QQR) = A000009 =    1    1    2    2    3    4    5    6    8    10   12   15   ...
A(QRP) = A279791 =    1    1    2    2    3    6    5    8    8    16   12   23   ...
A(QRQ) = A279791 =    1    1    2    2    3    6    5    8    8    16   12   23   ...
A(QRR) = A000012 =    1    1    1    1    1    1    1    1    1    1    1    1    ...

A(RPP) = A047968 =    1    3    4    8    8    17   16   30   34   52   57   99   ...
A(RPQ) = A000005 =    1    2    2    3    2    4    2    4    3    4    2    6    ...
A(RPR) = A047968 =    1    3    4    8    8    17   16   30   34   52   57   99   ...
A(RQP) = A047966 =    1    2    3    4    4    8    6    10   11   15   13   25   ...
A(RQQ) = A000012 =    1    1    1    1    1    1    1    1    1    1    1    1    ...
A(RQR) = A047966 =    1    2    3    4    4    8    6    10   11   15   13   25   ...
A(RRP) = A007425 =    1    3    3    6    3    9    3    10   6    9    3    18   ...
A(RRQ) = A000005 =    1    2    2    3    2    4    2    4    3    4    2    6    ...
A(RRR) = A007425 =    1    3    3    6    3    9    3    10   6    9    3    18   ...

With duplicates removed there are a total of 18 different sequences: A000005, A000009, A000012, A007425, A047966, A047968, A063834, A270995, A271619, A279375, A279784, A279785, A279786, A279787, A279788, A279789, A279790, A279791.