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

Author

Aidan Markey, Jan 12 2025

Keywords

• nonn
• changed

Examples

			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.
		

Links

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

Crossrefs

Similar to A380124, A380126.

Cf. A008284, A072233, A238312.

Formula

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).

Extensions

More terms from Chai Wah Wu, Feb 19 2025

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