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

0,4

Links

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

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).

Example

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).

Crossrefs

Similar to A380125, A380126.

Cf. A008284, A072233.

Sequence in context: A247374 A336512 A046994 * A058811 A101822 A354538

Adjacent sequences: A380121 A380122 A380123 * A380125 A380126 A380127

Keyword

nonn

Author

Aidan Markey, Jan 12 2025

Extensions

More terms from Chai Wah Wu, Feb 19 2025

Status

approved