%I #9 Apr 22 2022 10:36:51
%S 1,0,1,31,4,0,5474,292,9,0,2554091,72318,1206,10,0,2502018819,
%T 43707943,438987,2871,5,0,4456194509950,52717010017,351487598,1622954,
%U 4355,1,0,13077453070386914,111615599664989,528618296314,1764575884,4080889,4385,0,0
%N Triangle read by rows: T(n,k) is the number of set partitions of {1..4n} into n sets of 4 with k disjoint strings of adjacent sets, each being a contiguous set of elements
%C Number of configurations with k connected components (consisting of polyomino matchings) in the generalized game of memory played on the path of length 4n, see [Young].
%H Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.
%F G.f.: Sum_{j>=0} (4*j)! * y^j * (1-(1-z)*y)^(4*j+1) / (j! * 24^j * (1-(1-z)*y^2)^(4*j+1)).
%e Triangle begins:
%e 1;
%e 0, 1;
%e 31, 4, 0;
%e 5474, 292, 9, 0;
%e 2554091, 72318,1206, 10, 0;
%e ...
%e For n=2 and k=1 the configurations are (1,6,7,8),(2,3,4,5), as well as (1,2,7,8),(3,4,5,6) and also (1,2,3,8),(4,5,6,7) (i.e. configurations with a single contiguous set) and (1,2,3,4),(5,6,7,8) (i.e. two adjacent contiguous sets); hence T(2,1) = 4.
%t CoefficientList[Normal[Series[Sum[y^j*(4*j)!/24^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(4*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]
%o (PARI)
%o T(n)={my(v=Vec(sum(j=0, n, (4*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(4*j+1) / (j! * 24^j * (1-(1-y)*x^2 + O(x*x^n))^(4*j+1))))); vector(#v, i, Vecrev(v[i], i))}
%o { my(A=T(8)); for(n=1, #A, print(A[n])) }
%Y Row sums are A025036.
%Y Column k=0 is column 0 of A334057.
%Y Cf. A079267, A334056, A334057, A334058, A334059, A334060, A325753.
%K nonn,tabl
%O 0,4
%A _Donovan Young_, May 26 2020