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.
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
Examples
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.
Links
- 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.
Crossrefs
Programs
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Mathematica
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}];
Formula
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))).
Comments