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Number of unordered pairs of partitions of n with no common parts.
10

%I #12 Feb 07 2024 11:58:01

%S 1,0,1,2,6,8,24,30,74,110,219,309,651,870,1608,2394,4085,5756,9931,

%T 13785,22724,32300,50404,70862,111540,153756,232868,326259,484090,

%U 667015,986082,1345566,1951216,2673588,3805742,5179213,7348514,9895254,13845750,18681896

%N Number of unordered pairs of partitions of n with no common parts.

%H Reinhard Zumkeller, <a href="/A260669/b260669.txt">Table of n, a(n) for n = 0..5000</a>

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

%e 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:

%e . -------------------+---+---+---+---+---+---+---+---+---+---+---+

%e . | a | b | c | d | e | f | g | h | i | j | k |

%e . ---+---------------+---+---+---+---+---+---+---+---+---+---+---+

%e . a | [6] | 1 |

%e . b | [5,1] | 0 2 |

%e . c | [4,2] | 0 0 2 |

%e . d | [4,1,1] | 0 1 1 3 |

%e . e | [3,3] | 0 0 0 0 2 |

%e . f | [3,2,1] | 0 1 1 1 1 3 |

%e . g | [3,1,1,1] | 0 1 0 2 1 2 4 |

%e . h | [2,2,2] | 0 0 1 0 0 1 0 3 |

%e . i | [2,2,1,1] | 0 1 1 2 0 2 2 2 4 |

%e . j | [2,1,1,1,1] | 0 1 1 2 0 2 3 1 3 5 |

%e . k | [1,1,1,1,1,1] | 0 1 0 2 0 1 3 0 2 4 6 |

%e . ---+---------------+---+---+---+---+---+---+---+---+---+---+---+

%e The table contains 24 zeros, therefore a(6) = 24.

%o (Haskell)

%o a260669 = flip div 2 . a054440

%Y Cf. A000041, A054440.

%K nonn

%O 0,4

%A _Reinhard Zumkeller_, Nov 15 2015

%E a(0)=1 prepended by _Alois P. Heinz_, Feb 07 2024