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A318244 a(n) is the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that only one such pair is joined by an edge.

%I A318244 #29 Mar 18 2020 11:06:32
%S A318244 1,0,8,34,347,3666,47484,707480,11971341,226599568,4744010444,
%T A318244 108834109034,2714992695407,73169624071138,2118530753728184,
%U A318244 65582753432993648,2161565971116312537,75572040870327124064,2793429487732659591888,108847840347732886117874,4459207771645802095292995
%N A318244 a(n) is the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that only one such pair is joined by an edge.
%C A318244 This is a companion entry to A318243 and uses an inclusion-exclusion method on the matching numbers given there.
%C A318244 This is also the number of "1-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - _Donovan Young_, Oct 23 2018
%H A318244 D. Young, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Young/young2.html">The Number of Domino Matchings in the Game of Memory</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.
%H A318244 Donovan Young, <a href="https://arxiv.org/abs/1905.13165">Generating Functions for Domino Matchings in the 2 * k Game of Memory</a>, arXiv:1905.13165 [math.CO], 2019. Also in <a href="https://www.emis.de/journals/JIS/VOL22/Young/young13.html">J. Int. Seq.</a>, Vol. 22 (2019), Article 19.8.7.
%F A318244 a(n) = Sum_{k=0..n-1} (-1)^k*(2*n-2*k-3)!! * A318243(n,k) where and 0!! = (-1)!! = 1; proved by inclusion-exclusion.
%e A318244 For the case n = 2, if one pair is joined by an edge, then the remaining pair is forced to be joined by the remaining edge. Thus a(2) = 0.
%Y A318244 Cf. A046741, A318243, A318267, A318268, A318269, A318270. When no pair is joined by an edge, the number of configurations is given by A265167.
%K A318244 nonn
%O A318244 1,3
%A A318244 _Donovan Young_, Aug 22 2018