A325753 Triangle read by rows giving the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that exactly k such pairs are joined by an edge.

1, 0, 1, 1, 0, 2, 2, 8, 2, 3, 21, 34, 39, 6, 5, 186, 347, 250, 138, 16, 8, 2113, 3666, 2919, 1234, 414, 36, 13, 27856, 47484, 36714, 17050, 4830, 1104, 76, 21, 422481, 707480, 545788, 253386, 78815, 16174, 2715, 152, 34, 7241480, 11971341, 9195198, 4317996, 1369260, 309075, 48444, 6282, 294, 55

Offset: 0

Author

Donovan Young, May 18 2019

Keywords

• nonn
• tabl

Comments

This is the number of "k-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. First column is A265167, second column is A318244. Diagonals are given by A000045, A178523, A318267, A318268, A318269, A318270.

Examples

			The first few rows of T(n,k) are:
   1;
   0,  1;
   1,  0,  2;
   2,  8,  2,  3;
  21, 34, 39,  6, 5;
  ...
For n = 2 there is only one way to place the two pairs such that neither is joined by an edge, hence T(2,0)=1. If one pair is joined by an edge, the other is forced to be, hence T(2,1) = 0, and since the pairs can be joined horizontally or vertically T(2,2) = 2.
		

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, A178523, A265167, A318243, A318244, A318267, A318268, A318269, A318270.

Programs

Mathematica

CoefficientList[Normal[Series[Sum[Factorial2[2*k-1]*y^k*(1-(1-z)*y)^k/(1+(1-z)*y)^k/(1+(1-z)*y-(1-z)^2*y^2)^(k+1),{k,0,20}],{y,0,20}]],{y,z}];

Formula

G.f.: Sum_{j>=0} (2*j-1)!! * y^j * (1-(1-z)*y)^j / (1+(1-z)*y)^j / (1+(1-z)*y-(1-z)^2*y^2)^(j+1).