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%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
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