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 A384116 #9 May 20 2025 12:52:15
%S A384116 1,1,1,1,0,1,1,1,1,1,1,4,9,4,1,1,11,39,39,11,1,1,26,183,334,183,26,1,
%T A384116 1,57,833,3087,3087,833,57,1,1,120,3629,27472,53731,27472,3629,120,1,
%U A384116 1,247,15291,236127,922515,922515,236127,15291,247,1,1,502,63051,1975246,15524639,30844786,15524639,1975246,63051,502,1
%N A384116 Array read by antidiagonals: T(n,m) is the number of total dominating sets in the n X m rook graph K_n X K_m.
%H A384116 Andrew Howroyd, <a href="/A384116/b384116.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A384116 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%H A384116 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>.
%F A384116 T(n,m) = B(n,m) - Sum_{i=1..m} (-1)^i*binomial(m,i)*B(m-i,n), where B(n,m) = Sum_{i=0..m} (-1)^i*binomial(n,i)*binomial(m,i)*i!*(2^(n-i)-1)^(m-i).
%F A384116 T(n,m) = T(m,n).
%e A384116 Array begins:
%e A384116 =================================================================
%e A384116 n\m | 0 1 2 3 4 5 6 ...
%e A384116 ----+------------------------------------------------------------
%e A384116 0 | 1 1 1 1 1 1 1 ...
%e A384116 1 | 1 0 1 4 11 26 57 ...
%e A384116 2 | 1 1 9 39 183 833 3629 ...
%e A384116 3 | 1 4 39 334 3087 27472 236127 ...
%e A384116 4 | 1 11 183 3087 53731 922515 15524639 ...
%e A384116 5 | 1 26 833 27472 922515 30844786 1019569593 ...
%e A384116 6 | 1 57 3629 236127 15524639 1019569593 66544564805 ...
%e A384116 7 | 1 120 15291 1975246 256594143 33329148492 4314985562475 ...
%e A384116 ...
%o A384116 (PARI)
%o A384116 B(n,m) = {sum(i=0, min(n,m), (-1)^i*binomial(n,i)*binomial(m,i)*i!*(2^(n-i)-1)^(m-i))}
%o A384116 T(n,m) = {B(n,m) - sum(i=1, m, (-1)^i*binomial(m,i)*B(m-i,n))}
%Y A384116 Main diagonal is A303208.
%Y A384116 Column 0 is A000012.
%Y A384116 Column 1 is A000295(n), n > 0.
%Y A384116 Column 2 is A287063(n), n > 1.
%Y A384116 Cf. A287274, A384117, A384118.
%K A384116 nonn,tabl
%O A384116 0,12
%A A384116 _Andrew Howroyd_, May 19 2025