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A318243 Triangle read by rows giving the sum of the number of k-matchings of the graphs obtained by deleting one edge and its two vertices from the ladder graph L_n = P_2 X P_n in all possible ways. 7
1, 4, 4, 7, 22, 9, 10, 58, 78, 20, 13, 112, 282, 224, 40, 16, 184, 702, 1052, 570, 78, 19, 274, 1419, 3260, 3335, 1338, 147, 22, 382, 2514, 7928, 12520, 9462, 2968, 272, 25, 508, 4068, 16460, 35955, 42108, 24766, 6312, 495, 28, 652, 6162, 30584, 86330, 140586, 128352, 60976, 12996, 890, 31, 814, 8877, 52352 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

T(n,k) is useful for computing the number of configurations of n indistinguishable pairs placed on the vertices of P_2 X P_n such that only one such pair is joined by an edge. The g.f. given below can be proven using the calculus of the rook polynomial associated with A046741.

LINKS

Table of n, a(n) for n=1..59.

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

G.f.: (1 + 2*t*z + 2*z/(1-t*z)^2)*(1-t*z)^2/(1 - z - 2*t*z - t*z^2 + t^3*z^3)^2.

EXAMPLE

The first few rows of T(n,k) are

   1;

   4,   4;

   7,  22,   9;

  10,  58,  78,  20;

  13, 112, 282, 224,  40;

For n = 2, the four ways of deleting an edge and its vertices from P_2 X P_2 all yield a graph with two vertices joined by an edge. This graph has 1 0-matching and 1 1-matching, thus T(2,k) = 4, 4.

MATHEMATICA

CoefficientList[Normal[Series[(1 + 2*t*z + 2*z/(1-t*z)^2)*(1-t*z)^2/(1 - z - 2*t*z - t*z^2 + t^3*z^3)^2, {z, 0, 10}]], {z, t}]//MatrixForm

CROSSREFS

Cf. A046741, A318244, A318267, A318268, A318269, A318270.

Sequence in context: A330765 A336718 A173324 * A321661 A183492 A019159

Adjacent sequences:  A318240 A318241 A318242 * A318244 A318245 A318246

KEYWORD

nonn,tabl

AUTHOR

Donovan Young, Aug 22 2018

STATUS

approved

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Last modified August 6 18:20 EDT 2020. Contains 336256 sequences. (Running on oeis4.)