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

1,3

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

Links

Table of n, a(n) for n=1..21.

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.

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.

Example

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.

Crossrefs

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

Sequence in context: A223015 A222796 A203445 * A280395 A158991 A265161

Adjacent sequences: A318241 A318242 A318243 * A318245 A318246 A318247

Keyword

nonn

Author

Donovan Young, Aug 22 2018

Status

approved