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A325754
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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 a horizontal edge.
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0
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1, 1, 0, 2, 0, 1, 7, 4, 4, 0, 43, 38, 21, 2, 1, 372, 360, 168, 36, 9, 0, 4027, 3972, 1818, 478, 93, 6, 1, 51871, 51444, 23760, 6640, 1260, 144, 16, 0, 773186, 768732, 358723, 103154, 20205, 2734, 278, 12, 1, 13083385, 13027060, 6129670, 1796740, 363595, 52900, 5650, 400, 25, 0
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OFFSET
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0,4
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COMMENTS
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This is the number of "k-horizontal-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young].
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LINKS
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FORMULA
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G.f.: Sum_{j>=0} (2*j-1)!! y^j/(1-(1-z)*y)/(1+(1-z)*y)^(2*j+1).
E.g.f.: exp((sqrt(1 - 2 y)-1) (1 - z))/sqrt(1 - 2 y) - exp((y - 2) (1 - z)) sqrt(Pi/2) sqrt(1 - z) (-erfi(sqrt(2) sqrt(1 - z)) + erfi(((1 + sqrt(1 - 2 y)) sqrt(1 - z))/sqrt(2))).
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EXAMPLE
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The first few rows of T(n,k) are:
1;
1, 0;
2, 0, 1;
7, 4, 4, 0;
43, 38, 21, 2, 1;
...
For n=2, let the vertex set of P_2 X P_2 be {A,B,C,D} and the edge set be {AB, AC, BD, CD}, where AB and CD are horizontal edges. For k=0, we may place the pairs on A, C and B, D or on A, D and B, C, hence T(2,0) = 2. If we place a pair on one of the horizontal edges we are forced to place the other pair on the remaining horizontal edge, hence T(2,1)=0 and T(2,2)=1.
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MATHEMATICA
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CoefficientList[Normal[Series[Sum[Factorial2[2*k-1]*y^k/(1-(1-z)*y)/(1+(1-z)*y)^(2*k+1), {k, 0, 20}], {y, 0, 20}]], {y, z}];
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CROSSREFS
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Cf. A046741, A055140, A079267 A178523, A265167, A318243, A318244, A318267, A318268, A318269, A318270, A325753.
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KEYWORD
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AUTHOR
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STATUS
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approved
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