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 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. 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 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. LINKS Table of n, a(n) for n=0..54. 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.: 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). EXAMPLE 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. 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}]; CROSSREFS Cf. A046741, A178523, A265167, A318243, A318244, A318267, A318268, A318269, A318270. Sequence in context: A037300 A368877 A029623 * A208133 A046644 A343059 Adjacent sequences: A325750 A325751 A325752 * A325754 A325755 A325756 KEYWORD nonn,tabl AUTHOR Donovan Young, May 18 2019 STATUS approved

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Last modified August 11 22:08 EDT 2024. Contains 375076 sequences. (Running on oeis4.)