This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384117 #11 May 20 2025 12:52:11
%S A384117 1,1,1,1,0,1,1,1,1,1,1,3,4,3,1,1,6,3,3,6,1,1,10,4,6,4,10,1,1,15,5,4,4,
%T A384117 5,15,1,1,21,6,5,80,5,6,21,1,1,28,7,6,65,65,6,7,28,1,1,36,8,7,96,410,
%U A384117 96,7,8,36,1,1,45,9,8,133,306,306,133,8,9,45,1
%N A384117 Array read by antidiagonals: T(n,m) is the number of minimum total dominating sets in the n X m rook graph K_n X K_m.
%C A384117 For 1 < m <= n, the minimum size of a total dominating set is m. When 1 < m < n, solutions have exactly one vertex in each column. In the special case of n = m, solutions either have exactly one vertex in each column or have exactly one vertex in each row.
%H A384117 Andrew Howroyd, <a href="/A384117/b384117.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A384117 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumTotalDominatingSet.html">Minimum Total Dominating Set</a>.
%H A384117 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%F A384117 T(n,m) = Sum_{i=0..m} (-1)^i * binomial(m,i) * binomial(n,i) * i! * (n-i)^(m-i) for 1 < m < n.
%F A384117 T(n,m) = T(m,n).
%F A384117 T(n,2) = T(n,3) = n for n >= 4.
%e A384117 Array begins:
%e A384117 ============================================
%e A384117 n\m | 0 1 2 3 4 5 6 7 8 ...
%e A384117 ----+---------------------------------------
%e A384117 0 | 1 1 1 1 1 1 1 1 1 ...
%e A384117 1 | 1 0 1 3 6 10 15 21 28 ...
%e A384117 2 | 1 1 4 3 4 5 6 7 8 ...
%e A384117 3 | 1 3 3 6 4 5 6 7 8 ...
%e A384117 4 | 1 6 4 4 80 65 96 133 176 ...
%e A384117 5 | 1 10 5 5 65 410 306 427 568 ...
%e A384117 6 | 1 15 6 6 96 306 5112 4207 6448 ...
%e A384117 7 | 1 21 7 7 133 427 4207 48818 38424 ...
%e A384117 8 | 1 28 8 8 176 568 6448 38424 695424 ...
%e A384117 ...
%o A384117 (PARI)
%o A384117 B(n,k) = {if(k<=n, if(k==1, binomial(n,2), sum(i=0, k, (-1)^i * binomial(k,i) * binomial(n,i) * i! * (n-i)^(k-i))))}
%o A384117 T(n,m) = {if(n==0&&m==0, 1, B(n,m) + B(m,n))}
%Y A384117 Main diagonal is A303211.
%Y A384117 Column 0 is A000012.
%Y A384117 Column 1 is A000217(n-1), n > 0.
%Y A384117 Cf. A384116, A384118, A384119.
%K A384117 nonn,tabl
%O A384117 0,12
%A A384117 _Andrew Howroyd_, May 19 2025