A260669 Number of unordered pairs of partitions of n with no common parts.

1, 0, 1, 2, 6, 8, 24, 30, 74, 110, 219, 309, 651, 870, 1608, 2394, 4085, 5756, 9931, 13785, 22724, 32300, 50404, 70862, 111540, 153756, 232868, 326259, 484090, 667015, 986082, 1345566, 1951216, 2673588, 3805742, 5179213, 7348514, 9895254, 13845750, 18681896

Offset: 0

Author

Reinhard Zumkeller, Nov 15 2015

Keywords

• nonn

Examples

			n = 6 has A000041(6) = 11 partitions: [6], [5,1], [4,2], [4,1,1], [3,3], [3,2,1], [3,1,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1], [1,1,1,1,1,1]; the following table shows the number of common parts of the pairs of these partitions, e.g. row i, col f: number of common parts of [2,2,1,1] and [3,2,1] = 2:
. -------------------+---+---+---+---+---+---+---+---+---+---+---+
.                    | a | b | c | d | e | f | g | h | i | j | k |
. ---+---------------+---+---+---+---+---+---+---+---+---+---+---+
.  a | [6]           | 1                                         |
.  b | [5,1]         | 0   2                                     |
.  c | [4,2]         | 0   0   2                                 |
.  d | [4,1,1]       | 0   1   1   3                             |
.  e | [3,3]         | 0   0   0   0   2                         |
.  f | [3,2,1]       | 0   1   1   1   1   3                     |
.  g | [3,1,1,1]     | 0   1   0   2   1   2   4                 |
.  h | [2,2,2]       | 0   0   1   0   0   1   0   3             |
.  i | [2,2,1,1]     | 0   1   1   2   0   2   2   2   4         |
.  j | [2,1,1,1,1]   | 0   1   1   2   0   2   3   1   3   5     |
.  k | [1,1,1,1,1,1] | 0   1   0   2   0   1   3   0   2   4   6 |
. ---+---------------+---+---+---+---+---+---+---+---+---+---+---+
The table contains 24 zeros, therefore a(6) = 24.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000041, A054440.

Programs

Haskell

a260669 = flip div 2 . a054440

Formula

a(n) = A054440(n) / 2 for n >= 1.

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 07 2024