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%I #26 Mar 12 2024 15:46:28
%S 1,0,1,0,0,6,4,1,0,0,0,48,117,126,84,36,9,1,0,0,0,0,488,2640,6712,
%T 10864,12726,11424,8008,4368,1820,560,120,16,1,0,0,0,0,0,6130,58300,
%U 269500,808325,1778875,3075160,4349400,5154900,5186300,4454400,3268360,2042950,1081575,480700,177100,53130,12650,2300,300,25,1
%N Irregular triangle read by rows: T(n,k) is the number of dominating subsets with cardinality k of the n X n rook graph (n >= 1, 0 <= k <= n^2).
%C The entries in row n are the coefficients of the domination polynomial of the n X n rook graph.
%C Sum of entries in row n = A287065 = main diagonal of A287274.
%C Number of minimum dominating sets T(n,n) = A248744(n)
%D John J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, 2004, chapter 7.
%H Stephan Mertens, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Mertens/mertens6.html">Domination Polynomial of the Rook Graph</a>, Journal of Integer Sequences 27 (2024), Article 24.3.7; <a href="https://arxiv.org/abs/2401.00716">arXiv:2401.00716</a> [math.CO], 2024.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%F G.f.: ((x+1)^n - 1)^m - (-1)^m + Sum_{k=0..m-1} binomial(m,k)*(-1)^k*((1+x)^k - 1)^n (for the rectangular n X m rook graph).
%F T(n,n) = 2*n^n - n!.
%e Triangle begins: (first 5 rows)
%e 1;
%e 0,1;
%e 0,0,6,4,1;
%e 0,0,0,48,117,126,84,36,9,1;
%e 0,0,0,0,488,2640,6712,10864,12726,11424,8008,4368,1820,560,120,16,1;
%e ...
%t R[n_, m_] := CoefficientList[((x + 1)^n - 1)^m - (-1)^m*Sum[Binomial[m, k]*(-1)^k*((1 + x)^k - 1)^n, {k, 0, m - 1}], x];
%t Flatten[Table[R[n,n],{n,1,5}]
%Y Cf. A287065 (row sums), A287274, A248744 (leading diagonal).
%K nonn,tabf
%O 0,6
%A _Stephan Mertens_, Jan 07 2024
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