A380126 Total number of ways of partitioning n and any natural number less than or equal to n into the same number of parts, not treating partitions of n and itself in a different order as distinct.

1, 3, 6, 14, 26, 58, 106, 214, 394, 742, 1314, 2406, 4139, 7234, 12250, 20778, 34279, 56805, 91866, 148816, 236772, 375899, 588208, 919235, 1417538, 2180608, 3320197, 5038918, 7577850, 11363516, 16899942, 25056925, 36892553, 54136934, 78951553, 114783293, 165922204

Offset

1,2

Links

Table of n, a(n) for n=1..37.

Formula

a(n) = Sum_{i=1..n-1} Sum_{j=1..i} p(n,j)*p(i,j) + Sum_{j=1..n} (p(n,j)*(p(n,j)+1))/2, where p(n,j) is the number of partitions of n into j positive parts (A008284, A072233).

Example

For example, a(4)=14:
  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+2,2+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.

Crossrefs

Similar to A380124, A380125.

Cf. A008284, A072233.

Sequence in context: A358831 A323450 A051749 * A278788 A279828 A277954

Adjacent sequences: A380123 A380124 A380125 * A380128 A380129 A380130

Keyword

nonn,changed

Author

Aidan Markey, Jan 12 2025

Extensions

More terms from Chai Wah Wu, Feb 19 2025

Status

approved