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%I #24 Jan 07 2024 15:29:30
%S 1,0,1,121,4,1,124760,1347,18,1,486854621,2001548,8154,52,1,
%T 5184423824705,10231953233,17045774,35542,121,1,123243726413573515,
%U 134835947255262,112619668659,102416812,124881,246,1,5717986519188343198259,3821094862609800013,1820735766620673,863827126967,486979381,375627,455,1
%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 5n.
%C In this generalized game of memory n indistinguishable quintuples of matched cards are placed on the vertices of the path of length 5n. A polyomino is a quintuple on five adjacent vertices.
%C T(n,k) is the number of set partitions of {1..5n} into n sets of 5 with k of the sets being a contiguous set of elements. - _Andrew Howroyd_, Apr 16 2020
%H Andrew Howroyd, <a href="/A334058/b334058.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} (5*j)! * y^j / (j! * 120^j * (1+(1-z)*y)^(5*j+1)).
%F T(n,k) = Sum_{j=0..n-k} (-1)^(n-j-k)*(n+4*j)!/(120^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 121, 4, 1;
%e 124760, 1347, 18, 1;
%e 486854621, 2001548, 8154, 52, 1;
%e ...
%e For n=2 and k=1 the polyomino must start either on the second, third, fourth, or fifth vertex of the path, otherwise the remaining quintuple will also form a polyomino; thus T(2,1) = 4.
%t CoefficientList[Normal[Series[Sum[y^j*(5*j)!/120^j/j!/(1+y*(1-z))^(5*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+4*j)!/(120^j*j!*(n-j-k)!*k!))} \\ _Andrew Howroyd_, Apr 16 2020
%Y Row sums are A025037.
%Y Cf. A079267, A334056, A334057, A325753.
%K nonn,tabl
%O 0,4
%A _Donovan Young_, Apr 15 2020
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