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A334058 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.

%I A334058 #24 Jan 07 2024 15:29:30
%S A334058 1,0,1,121,4,1,124760,1347,18,1,486854621,2001548,8154,52,1,
%T A334058 5184423824705,10231953233,17045774,35542,121,1,123243726413573515,
%U A334058 134835947255262,112619668659,102416812,124881,246,1,5717986519188343198259,3821094862609800013,1820735766620673,863827126967,486979381,375627,455,1
%N A334058 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 A334058 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 A334058 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 A334058 Andrew Howroyd, <a href="/A334058/b334058.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A334058 Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.
%F A334058 G.f.: Sum_{j>=0} (5*j)! * y^j / (j! * 120^j * (1+(1-z)*y)^(5*j+1)).
%F A334058 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 A334058 The first few rows of T(n,k) are:
%e A334058           1;
%e A334058           0,       1;
%e A334058         121,       4,    1;
%e A334058      124760,    1347,   18,  1;
%e A334058   486854621, 2001548, 8154, 52, 1;
%e A334058   ...
%e A334058 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 A334058 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 A334058 (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 A334058 Row sums are A025037.
%Y A334058 Cf. A079267, A334056, A334057, A325753.
%K A334058 nonn,tabl
%O A334058 0,4
%A A334058 _Donovan Young_, Apr 15 2020