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.
Original entry on oeis.org
1, 0, 8, 34, 347, 3666, 47484, 707480, 11971341, 226599568, 4744010444, 108834109034, 2714992695407, 73169624071138, 2118530753728184, 65582753432993648, 2161565971116312537, 75572040870327124064, 2793429487732659591888, 108847840347732886117874, 4459207771645802095292995
Offset: 1
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.
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.
Original entry on oeis.org
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
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.
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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}];
A325754
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.
Original entry on oeis.org
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
Offset: 0
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.
Cf.
A046741,
A055140,
A079267 A178523,
A265167,
A318243,
A318244,
A318267,
A318268,
A318269,
A318270,
A325753.
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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}];
Showing 1-3 of 3 results.
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