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A334057 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.

%I A334057 #30 Jan 07 2024 15:33:26
%S A334057 1,0,1,31,3,1,5474,288,12,1,2554091,72026,1476,31,1,2502018819,
%T A334057 43635625,508610,5505,65,1,4456194509950,52673302074,394246455,
%U A334057 2559565,16710,120,1,13077453070386914,111562882654972,580589062179,2504572910,10288390,43806,203,1
%N A334057 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 A334057 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 A334057 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 A334057 Andrew Howroyd, <a href="/A334057/b334057.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A334057 Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.
%F A334057 G.f.: Sum_{j>=0} (4*j)! * y^j / (j! * 24^j * (1+(1-z)*y)^(4*j+1)).
%F A334057 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 A334057 The first few rows of T(n,k) are:
%e A334057         1;
%e A334057         0,     1;
%e A334057        31,     3,    1;
%e A334057      5474,   288,   12,  1;
%e A334057   2554091, 72026, 1476, 31, 1;
%e A334057   ...
%e A334057 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 A334057 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 A334057 (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 A334057 Row sums are A025036.
%Y A334057 Cf. A079267, A334056, A334058, A325753.
%K A334057 nonn,tabl
%O A334057 0,4
%A A334057 _Donovan Young_, Apr 15 2020