

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
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OFFSET

0,6


COMMENTS



LINKS



FORMULA

G.f.: Sum_{j>=0} (2*j1)!! * y^j * (1(1z)*y)^j / (1+(1z)*y)^j / (1+(1z)*y(1z)^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*k1]*y^k*(1(1z)*y)^k/(1+(1z)*y)^k/(1+(1z)*y(1z)^2*y^2)^(k+1), {k, 0, 20}], {y, 0, 20}]], {y, z}];


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



