OFFSET
0,4
COMMENTS
Number of configurations with k connected components (consisting of polyomino matchings) in the generalized game of memory played on the path of length 3n, see [Young].
LINKS
Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.
FORMULA
G.f.: Sum_{j>=0} (3*j)! * y^j * (1-(1-z)*y)^(3*j+1) / (j! * 6^j * (1-(1-z)*y^2)^(3*j+1)).
EXAMPLE
Triangle begins:
1;
0, 1;
7, 3, 0;
219, 56, 5, 0;
12861, 2352, 183, 4, 0;
...
For n=2 and k=1 the configurations are (1,5,6),(2,3,4) and (1,2,6),(3,4,5) (i.e. configurations with a single contiguous set) and (1,2,3),(4,5,6) (i.e. two adjacent contiguous sets); hence T(2,1) = 3.
MATHEMATICA
CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(3*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]
PROG
(PARI)
T(n)={my(v=Vec(sum(j=0, n, (3*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(3*j+1) / (j! * 6^j * (1-(1-y)*x^2 + O(x*x^n))^(3*j+1))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(8)); for(n=1, #A, print(A[n])) }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Donovan Young, May 26 2020
STATUS
approved