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.

1, 0, 8, 34, 347, 3666, 47484, 707480, 11971341, 226599568, 4744010444, 108834109034, 2714992695407, 73169624071138, 2118530753728184, 65582753432993648, 2161565971116312537, 75572040870327124064, 2793429487732659591888, 108847840347732886117874, 4459207771645802095292995

Offset: 1

Author

Donovan Young, Aug 22 2018

Keywords

• nonn

Comments

This is a companion entry to A318243 and uses an inclusion-exclusion method on the matching numbers given there.

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

Examples

			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.
		

Links

• D. Young, The Number of Domino Matchings in the Game of Memory, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.
• Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.

Crossrefs

Cf. A046741, A318243, A318267, A318268, A318269, A318270. When no pair is joined by an edge, the number of configurations is given by A265167.

Formula

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.