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

0,4

Links

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

Formula

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

Example

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.

Prog

(Haskell)
a260669 = flip div 2 . a054440

Crossrefs

Cf. A000041, A054440.

Sequence in context: A081957 A334901 A279732 * A122758 A122756 A193946

Adjacent sequences: A260666 A260667 A260668 * A260670 A260671 A260672

Keyword

nonn

Author

Reinhard Zumkeller, Nov 15 2015

Extensions

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

Status

approved