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A378561 Number of ways to place k nonattacking anassas on an n X n chess board. Triangle T(n,k) read by rows.

%I A378561 #35 Dec 28 2024 10:19:49
%S A378561 1,1,1,1,4,3,1,9,22,14,1,16,82,156,90,1,25,220,840,1366,738,1,36,485,
%T A378561 3100,9796,14288,7364,1,49,938,9030,46816,129360,174112,86608,1,64,
%U A378561 1652,22344,172116,767424,1916776,2424880,1173240,1,81,2712,49056,525756,3442740,13682320,31572720,38019496,17990600
%N A378561 Number of ways to place k nonattacking anassas on an n X n chess board. Triangle T(n,k) read by rows.
%C A378561 Anassas (also called semi-rook+semi-bishop) are chess pieces with 2 moves: one horizontal or vertical and one diagonal.
%H A378561 S. Chaiken, C. R. H. Hanusa, and T. Zaslavsky, <a href="https://jkms.kms.or.kr/journal/view.html?doi=10.4134/JKMS.j190682">A q-Queens Problem. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders</a>, J. Korean Math. Soc., 57(6): 1407-1433, 2020; see also <a href="https://arxiv.org/abs/1609.00853">arXiv preprint</a>, arXiv:1609.00853 [math.CO], 2016-2020.
%H A378561 Vaclav Kotesovec, <a href="http://www.kotesovec.cz/books/kotesovec_non_attacking_chess_pieces_2013_6ed.pdf">Non-attacking chess pieces</a>, 6ed, 2013, p. 716-722.
%H A378561 Eder G. Santos, <a href="https://arxiv.org/abs/2411.16492">Counting non-attacking chess pieces placements: Bishops and Anassas</a>, arXiv:2411.16492 [math.CO], 2024.
%F A378561 T(n,k) = Sum_{j=0..ceiling(k/2)} j! * binomial(n-k+j,j) * Stirling2(n,n-k+j) * 2^(k-2*j) * (binomial(k-j,j-1) + binomial(k-j+1,j)).
%e A378561 Triangle begins:
%e A378561   1;
%e A378561   1  1;
%e A378561   1  4   3;
%e A378561   1  9  22   14;
%e A378561   1 16  82  156   90;
%e A378561   1 25 220  840 1366   738;
%e A378561   1 36 485 3100 9796 14288 7364;
%e A378561   ...
%o A378561 (SageMath) print([sum([factorial(j)*binomial(n-k+j,j)*stirling_number2(n,n-k+j)*2^(k-2*j)*(binomial(k-j,j-1)+binomial(k-j+1,j)) for j in [0..ceil(k/2)]]) for n in [0..10] for k in [0..n]])
%Y A378561 Columns k=0-1 give: A000012, A000290.
%Y A378561 Main diagonal gives A088789(n+1).
%K A378561 nonn,easy,tabl
%O A378561 0,5
%A A378561 _Eder G. Santos_, Nov 30 2024