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

%I A260669 #12 Feb 07 2024 11:58:01
%S A260669 1,0,1,2,6,8,24,30,74,110,219,309,651,870,1608,2394,4085,5756,9931,
%T A260669 13785,22724,32300,50404,70862,111540,153756,232868,326259,484090,
%U A260669 667015,986082,1345566,1951216,2673588,3805742,5179213,7348514,9895254,13845750,18681896
%N A260669 Number of unordered pairs of partitions of n with no common parts.
%H A260669 Reinhard Zumkeller, <a href="/A260669/b260669.txt">Table of n, a(n) for n = 0..5000</a>
%F A260669 a(n) = A054440(n) / 2 for n >= 1.
%e A260669 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 A260669 . -------------------+---+---+---+---+---+---+---+---+---+---+---+
%e A260669 .                    | a | b | c | d | e | f | g | h | i | j | k |
%e A260669 . ---+---------------+---+---+---+---+---+---+---+---+---+---+---+
%e A260669 .  a | [6]           | 1                                         |
%e A260669 .  b | [5,1]         | 0   2                                     |
%e A260669 .  c | [4,2]         | 0   0   2                                 |
%e A260669 .  d | [4,1,1]       | 0   1   1   3                             |
%e A260669 .  e | [3,3]         | 0   0   0   0   2                         |
%e A260669 .  f | [3,2,1]       | 0   1   1   1   1   3                     |
%e A260669 .  g | [3,1,1,1]     | 0   1   0   2   1   2   4                 |
%e A260669 .  h | [2,2,2]       | 0   0   1   0   0   1   0   3             |
%e A260669 .  i | [2,2,1,1]     | 0   1   1   2   0   2   2   2   4         |
%e A260669 .  j | [2,1,1,1,1]   | 0   1   1   2   0   2   3   1   3   5     |
%e A260669 .  k | [1,1,1,1,1,1] | 0   1   0   2   0   1   3   0   2   4   6 |
%e A260669 . ---+---------------+---+---+---+---+---+---+---+---+---+---+---+
%e A260669 The table contains 24 zeros, therefore a(6) = 24.
%o A260669 (Haskell)
%o A260669 a260669 = flip div 2 . a054440
%Y A260669 Cf. A000041, A054440.
%K A260669 nonn
%O A260669 0,4
%A A260669 _Reinhard Zumkeller_, Nov 15 2015
%E A260669 a(0)=1 prepended by _Alois P. Heinz_, Feb 07 2024