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A334056 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 3n.

%I A334056 #43 Jan 07 2024 15:33:53
%S A334056 1,0,1,7,2,1,219,53,7,1,12861,2296,226,16,1,1215794,171785,13080,710,
%T A334056 30,1,169509845,19796274,1228655,53740,1835,50,1,32774737463,
%U A334056 3260279603,170725639,6250755,178325,4137,77,1,8400108766161,727564783392,32944247308,1036855344,25359670,507584,8428,112,1
%N A334056 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 3n.
%C A334056 In this generalized game of memory n indistinguishable triples of matched cards are placed on the vertices of the path of length 3n. A polyomino is a triple on three adjacent vertices. For dominoes in ordinary memory on the path of length 2n, see A079267.
%C A334056 T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k of the sets being a contiguous set of elements. - _Andrew Howroyd_, Apr 16 2020
%H A334056 Andrew Howroyd, <a href="/A334056/b334056.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A334056 Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Young/young5.html">J. Int. Seq.</a>, Vol. 23 (2020), Article 20.9.1.
%F A334056 G.f.: Sum_{j>=0} (3*j)! * y^j / (j! * 6^j * (1+(1-z)*y)^(3*j+1)).
%F A334056 T(n,k) = Sum_{j=0..n-k} (-1)^(n-j-k)*(n+2*j)!/(6^j*j!*(n-j-k)!*k!). - _Andrew Howroyd_, Apr 16 2020
%e A334056 The first few rows of T(n,k) are:
%e A334056       1;
%e A334056       0,    1;
%e A334056       7,    2,   1;
%e A334056     219,   53,   7,  1;
%e A334056   12861, 2296, 226, 16, 1;
%e A334056   ...
%e A334056 For n=2 and k=1 the polyomino must start either on the second vertex of the path, or the third, otherwise the remaining triple will also form a polyomino; thus T(2,1) = 2.
%t A334056 CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!/(1+y*(1-z))^(3*j+1),{j,0,20}],{y,0,20}]],{y,z}]
%o A334056 (PARI) T(n,k)={sum(j=0, n-k, (-1)^(n-j-k)*(n+2*j)!/(6^j*j!*(n-j-k)!*k!))} \\ _Andrew Howroyd_, Apr 16 2020
%Y A334056 Row sums are A025035.
%Y A334056 Cf. A079267, A334057, A334058, A325753.
%K A334056 nonn,tabl
%O A334056 0,4
%A A334056 _Donovan Young_, Apr 15 2020