A238312 -id:A238312 - cp's OEIS frontend

A380125 Total number of ways of partitioning n and any natural number less than or equal to n into the same number of parts, treating partitions of n and itself in a different order as distinct.

1, 1, 3, 6, 15, 28, 65, 119, 244, 450, 851, 1504, 2760, 4732, 8266, 13958, 23642, 38886, 64339, 103755, 167785, 266295, 422014, 658875, 1027992, 1581983, 2429719, 3692762, 5595987, 8401561, 12581456, 18682756, 27664577, 40675705, 59616335, 86831979, 126099127, 182065162

Offset: 0

Aidan Markey, Jan 12 2025

			For example, a(4)=15:
  4 and 1: (4,1),
  4 and 2: (4,2) (3+1,1+1) (2+2,1+1),
  4 and 3: (4,3) (3+1,2+1) (2+2,2+1) (2+1+1,1+1+1),
  4 and 4: (4,4) (3+1,3+1) (3+1,2+2) (2+2,3+1) (2+2,2+2) (2+1+1,2+1+1) (1+1+1+1,1+1+1+1).
Note that (3+1,2+2) and (2+2,3+1) are both counted.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Similar to A380124, A380126.

Cf. A008284, A072233, A238312.

a(n) = Sum_{i=0..n} Sum_{j=0..i} p(n,j)*p(i,j), where p(n,j) is the number of partitions of n into j positive parts (A008284, A072233).

a(n) = A380124(n) + A238312(n).

More terms from Chai Wah Wu, Feb 19 2025

a(0)=1 prepended by Alois P. Heinz, Aug 24 2025

A387259 Number of unordered pairs of partitions of n with the same number of parts.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 18, 28, 52, 86, 151, 246, 431, 694, 1167, 1884, 3095, 4904, 7919, 12379, 19596, 30315, 47117, 71922, 110332, 166403, 251547, 375575, 560787, 828276, 1223544, 1789656, 2616001, 3793295, 5491711, 7895309, 11333811, 16164595, 23013551, 32584816

Offset: 0

Aidan Markey, Aug 24 2025

The corresponding sequence for ordered pairs of partitions is A238312.

			For example, a(4)=6: (4,4) (3+1,3+1) (3+1,2+2) (2+2,2+2) (2+1+1,2+1+1) (1+1+1+1,1+1+1+1).
Note that (3+1,2+2) and (2+2,3+1) are not both counted.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A380124, A380126, A008284, A072233, A238312.

a(n) = Sum_{k=0..n} A000217(A072233(n,k)).

a(n) = A380126(n) - A380124(n).

cp's OEIS Frontend

A380125 Total number of ways of partitioning n and any natural number less than or equal to n into the same number of parts, treating partitions of n and itself in a different order as distinct.

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