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Number T(n,k) of ways to place k nonattacking queens on an n X n board; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
11

%I #23 Oct 07 2021 18:07:13

%S 1,1,1,1,4,0,1,9,8,0,1,16,44,24,2,1,25,140,204,82,10,1,36,340,1024,

%T 982,248,4,1,49,700,3628,7002,4618,832,40,1,64,1288,10320,34568,46736,

%U 22708,3192,92,1,81,2184,25096,131248,310496,312956,119180,13848,352,1,100,3480,54400,412596,1535440,2716096,2119176,636524,56832,724

%N Number T(n,k) of ways to place k nonattacking queens on an n X n board; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%e T(3,2) = 8:

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

%e |Q . .| |Q . .| |. . Q| |. . Q| |. . .| |. Q .| |. Q .| |. . .|

%e |. . Q| |. . .| |. . .| |Q . .| |Q . .| |. . .| |. . .| |. . Q|

%e |. . .| |. Q .| |. Q .| |. . .| |. . Q| |. . Q| |Q . .| |Q . .|

%e `-----´ `-----´ `-----´ `-----´ `-----´ `-----´ `-----´ `-----´.

%e Triangle T(n,k) begins:

%e 1;

%e 1, 1;

%e 1, 4, 0;

%e 1, 9, 8, 0;

%e 1, 16, 44, 24, 2;

%e 1, 25, 140, 204, 82, 10;

%e 1, 36, 340, 1024, 982, 248, 4;

%e 1, 49, 700, 3628, 7002, 4618, 832, 40;

%e 1, 64, 1288, 10320, 34568, 46736, 22708, 3192, 92;

%e 1, 81, 2184, 25096, 131248, 310496, 312956, 119180, 13848, 352;

%e ...

%Y Columns k=0-8 give: A000012, A000290, A036464, A047659, A061994, A108792, A176186, A178721, A252593.

%Y Main diagonal gives A000170.

%Y Row sums give A287227.

%Y T(2n,n) gives A348130.

%Y Cf. A090642, A144084, A178717.

%K nonn,tabl,hard

%O 0,5

%A _Alois P. Heinz_, Oct 01 2021