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Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 4n.
6

%I #30 Jan 07 2024 15:33:26

%S 1,0,1,31,3,1,5474,288,12,1,2554091,72026,1476,31,1,2502018819,

%T 43635625,508610,5505,65,1,4456194509950,52673302074,394246455,

%U 2559565,16710,120,1,13077453070386914,111562882654972,580589062179,2504572910,10288390,43806,203,1

%N Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 4n.

%C In this generalized game of memory n indistinguishable quadruples of matched cards are placed on the vertices of the path of length 4n. A polyomino is a quadruple on four adjacent vertices.

%C T(n,k) is the number of set partitions of {1..4n} into n sets of 4 with k of the sets being a contiguous set of elements. - _Andrew Howroyd_, Apr 16 2020

%H Andrew Howroyd, <a href="/A334057/b334057.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%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 / (j! * 24^j * (1+(1-z)*y)^(4*j+1)).

%F T(n,k) = Sum_{j=0..n-k} (-1)^(n-j-k)*(n+3*j)!/(24^j*j!*(n-j-k)!*k!). - _Andrew Howroyd_, Apr 16 2020

%e The first few rows of T(n,k) are:

%e 1;

%e 0, 1;

%e 31, 3, 1;

%e 5474, 288, 12, 1;

%e 2554091, 72026, 1476, 31, 1;

%e ...

%e For n=2 and k=1 the polyomino must start either on the second vertex of the path, the third, or the fourth, otherwise the remaining quadruple will also form a polyomino; thus T(2,1) = 3.

%t CoefficientList[Normal[Series[Sum[y^j*(4*j)!/24^j/j!/(1+y*(1-z))^(4*j+1),{j,0,20}],{y,0,20}]],{y,z}]

%o (PARI) T(n,k)={sum(j=0, n-k, (-1)^(n-j-k)*(n+3*j)!/(24^j*j!*(n-j-k)!*k!))} \\ _Andrew Howroyd_, Apr 16 2020

%Y Row sums are A025036.

%Y Cf. A079267, A334056, A334058, A325753.

%K nonn,tabl

%O 0,4

%A _Donovan Young_, Apr 15 2020