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A342372 Triangle T(n,k) of number of ways of arranging q nonattacking semi-queens on an n X n toroidal board, where 0 <= k <= n.

%I A342372 #34 Apr 07 2021 09:30:38
%S A342372 1,1,1,1,4,0,1,9,9,3,1,16,48,32,0,1,25,150,250,75,15,1,36,360,1200,
%T A342372 1224,288,0,1,49,735,4165,8869,6321,931,133,1,64,1344,11648,43136,
%U A342372 64512,33024,4096,0,1,81,2268,27972,160866,423306,469800
%N A342372 Triangle T(n,k) of number of ways of arranging q nonattacking semi-queens on an n X n toroidal board, where 0 <= k <= n.
%C A342372 T(0,0):=1 for combinatorial reasons.
%C A342372 A semi-queen can only move horizontal, vertical and parallel to the main diagonal of the board. Moves parallel to the secondary diagonal are not allowed.
%C A342372 Instead of a board on a torus, you can imagine that the semi-queens can leave a flat board on one side and re-enter the board on the other side.
%H A342372 Walter Trump, <a href="/A342372/b342372.txt">Table of n, a(n) for n = 1..222</a>
%H A342372 Walter Trump, <a href="/A342372/a342372_2.pdf">Semi-queen problem</a>
%F A342372 T(n,0) = 1.
%F A342372 T(n,1) = n^2.
%F A342372 T(n,2) = n^2*(n-1)*(n-2)/2.
%F A342372 T(n,3) = n^2*(n-1)*(n-2)*(n^2-6n+10)/6.
%F A342372 T(2n+1,2n+1) = A006717(n).
%F A342372 T(2n,2n) = 0.
%e A342372   1;
%e A342372   1,  1;
%e A342372   1,  4,   0;
%e A342372   1,  9,   9,   3;
%e A342372   1, 16,  48,  32,  0;
%e A342372   1, 25, 150, 250, 75, 15;
%Y A342372 Cf. A006717, A099152, A103220, A202654, A202655, A202656, A202657.
%K A342372 tabl,nonn
%O A342372 1,5
%A A342372 _Walter Trump_, Mar 09 2021