cp's OEIS Frontend

A303208 Number of total dominating sets in the n X n rook graph.

%I A303208 #18 May 22 2025 22:34:21
%S A303208 0,9,334,53731,30844786,66544564805,556588617042914,
%T A303208 18376877842518517955,2414913046805958120844234,
%U A303208 1267171440764716263069641387581,2658150749788131925244338204731596650,22299981643440069703358952237798936248817875
%N A303208 Number of total dominating sets in the n X n rook graph.
%H A303208 Andrew Howroyd, <a href="/A303208/b303208.txt">Table of n, a(n) for n = 1..50</a>
%H A303208 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%H A303208 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>
%F A303208 a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)^2*k!*A287065(n-k). - _Andrew Howroyd_, Apr 20 2018
%F A303208 a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Apr 20 2018
%t A303208 b[0] = 1; b[n_] := (2^n - 1)^n + Sum[Binomial[n, i] Sum[(-1)^j (-1 + 2^(n - j))^i Binomial[n, j], {j, 0, n}], {i, n - 1}]; Table[Sum[(-1)^k Binomial[n, k]^2 k! b[n - k], {k, 0, n}], {n, 10}]
%o A303208 (PARI) \\ here c(n) is A287065.
%o A303208 b(m, n)=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
%o A303208 c(n)=(2^n-1)^n + sum(i=1, n-1, b(n, i)*binomial(n, i));
%o A303208 a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)^2*k!*c(n-k))} \\ _Andrew Howroyd_, Apr 20 2018
%Y A303208 Main diagonal of A384116.
%Y A303208 Cf. A287065, A289196, A303211, A347921.
%K A303208 nonn
%O A303208 1,2
%A A303208 _Eric W. Weisstein_, Apr 19 2018
%E A303208 Terms a(6) and beyond from _Andrew Howroyd_, Apr 20 2018