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 A287274 #29 Feb 16 2025 08:33:46
%S A287274 1,3,3,7,11,7,15,51,51,15,31,227,421,227,31,63,963,3615,3615,963,63,
%T A287274 127,3971,30517,59747,30517,3971,127,255,16131,252231,989295,989295,
%U A287274 252231,16131,255,511,65027,2054941,16219187,32260381,16219187,2054941,65027,511
%N A287274 Array read by antidiagonals: T(m,n) = number of dominating sets in the lattice (rook) graph K_m X K_n.
%C A287274 A set of vertices can be represented as an m X n binary matrix. If all rows contain at least one 1 then regardless of what is in each row the set will form a dominating set giving (2^n-1)^m solutions. Otherwise if only i<m rows contain at least one 1 then all columns must contain a 1 for the set to form a dominating set giving A183109(i,n) solutions.
%H A287274 Andrew Howroyd, <a href="/A287274/b287274.txt">Table of n, a(n) for n = 1..780</a>
%H A287274 Stephan Mertens, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Mertens/mertens6.html">Domination Polynomial of the Rook Graph</a>, JIS 27 (2024) 24.3.7; <a href="https://arxiv.org/abs/2401.00716">arXiv:2401.00716</a> [math.CO], 2024.
%H A287274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>
%H A287274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%F A287274 T(m, n) = (2^n-1)^m + Sum_{i=1..m-1} binomial(m,i) * A183109(i,n).
%e A287274 Array begins:
%e A287274 =============================================================================
%e A287274 m\n| 1 2 3 4 5 6 7
%e A287274 ---|-------------------------------------------------------------------------
%e A287274 1 | 1 3 7 15 31 63 127...
%e A287274 2 | 3 11 51 227 963 3971 16131...
%e A287274 3 | 7 51 421 3615 30517 252231 2054941...
%e A287274 4 | 15 227 3615 59747 989295 16219187 263425695...
%e A287274 5 | 31 963 30517 989295 32260381 1048220463 33884452717...
%e A287274 6 | 63 3971 252231 16219187 1048220463 67680006971 4358402146791...
%e A287274 7 | 127 16131 2054941 263425695 33884452717 4358402146791 559876911043381...
%e A287274 ...
%t A287274 b[m_, n_] := Sum[(-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}];
%t A287274 a[m_, n_] := (2^n - 1)^m + Sum[ b[i, n]*Binomial[m, i], {i, 1, m - 1}];
%t A287274 Table[a[m - n + 1, n], {m, 1, 9}, {n, 1, m}] // Flatten (* _Jean-François Alcover_, Jun 12 2017, adapted from PARI *)
%o A287274 (PARI)
%o A287274 b(m,n)=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
%o A287274 a(m,n)=(2^n-1)^m + sum(i=1,m-1,b(i,n)*binomial(m,i));
%o A287274 for (i=1,7,for(j=1,7, print1(a(i,j), ",")); print);
%Y A287274 Main diagonal is A287065.
%Y A287274 Row 2 is A191341.
%Y A287274 Cf. A183109, A088699 (independent vertex sets), A290632.
%K A287274 nonn,tabl
%O A287274 1,2
%A A287274 _Andrew Howroyd_, May 22 2017