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, 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

1,2

Links

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

Formula

a(n) = Sum_{i=1..n} Sum_{j=1..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).

Example

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.

Crossrefs

Similar to A380124, A380126.

Cf. A008284, A072233, A238312.

Sequence in context: A285563 A285543 A318396 * A034953 A086737 A063834

Adjacent sequences: A380122 A380123 A380124 * A380126 A380127 A380128

Keyword

nonn

Author

Aidan Markey, Jan 12 2025

Extensions

More terms from Chai Wah Wu, Feb 19 2025

Status

approved