%I
%S 1,3,43,2619,654811,662827803,2699483026843,44102911693372059,
%T 2886238576935227688091,756075355087132847491422363,
%U 792522435884210281153847457333403,3323493099535510709729189614466101940379,55754039618636998102358059592995073452269940891
%N Number of winnable configurations in Lights Out game (played on a digraph) summed over every labeled digraph on n nodes.
%C Here a digraph may have at most one self loop (cf. A002416). A winnable configuration is a subset of lit vertices that can be turned off by some toggling sequence. In this version of the game, the digraph D is not necessarily symmetric so that the number of winnable configurations is 2^rank(A^t) where A^t is the transpose of the adjacency matrix of D.
%C In the limit as n goes to infinity, the probability that a random configuration on a random digraph is winnable is: Sum_{j>=0} (1/2^j) * (Product_{i>=j+1} (12^i))/(Product_{i>=1} (2^i  2^(ji))) = 0.610321...
%H A. Giffen and D. Parker, <a href="https://www.researchgate.net/publication/289241673_On_Generalizing_the_Lights_Out_Game_and_a_Generalization_of_Parity_Domination">On Generalizing the Lights Out Game and a Generalization of Parity Domination</a>, 2009.
%H L. Keough and D. Parker, <a href="https://arxiv.org/abs/1908.03649">An Extremal Problem for the Neighborhood Lights Out Game</a>, arXiv:1908.03649 [math.CO], 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LightsOutPuzzle.html">Lights Out Puzzle</a>
%F a(n) = Sum_{k=0..n} A286331(n,k)*2^k.
%F a(n) ~ c * 2^(n*(n+1)), where c = 0.610321518048266425924048782090628564983520109965690835927574616905934...  _Vaclav Kotesovec_, Apr 07 2020
%t Table[Table[2^k*Product[(2^n  2^i)^2 /(2^k  2^i), {i, 0, k  1}], {k, 0, n}] // Total, {n, 0, 12}]
%K nonn
%O 0,2
%A _Geoffrey Critzer_, Mar 15 2020
