A380124 Total number of ways of partitioning n and any natural number less than n into the same number of parts.

0, 0, 1, 3, 8, 17, 40, 78, 162, 308, 591, 1068, 1975, 3445, 6067, 10366, 17683, 29375, 48886, 79487, 129220, 206457, 328782, 516286, 808903, 1251135, 1929061, 2944622, 4478131, 6749574, 10139972, 15110286, 22440924, 33099258, 48645223, 71056244, 103449482, 149757609

Offset: 0

Author

Aidan Markey, Jan 12 2025

Keywords

• nonn
• changed

Examples

			For example, a(4)=8:
  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).
		

Links

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

Crossrefs

Similar to A380125, A380126.

Cf. A008284, A072233.

Formula

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

Extensions

More terms from Chai Wah Wu, Feb 19 2025