A334056 - OEIS

%I #43 Jan 07 2024 15:33:53

%S 1,0,1,7,2,1,219,53,7,1,12861,2296,226,16,1,1215794,171785,13080,710,

%T 30,1,169509845,19796274,1228655,53740,1835,50,1,32774737463,

%U 3260279603,170725639,6250755,178325,4137,77,1,8400108766161,727564783392,32944247308,1036855344,25359670,507584,8428,112,1

%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 3n.

%C 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 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 Andrew Howroyd, <a href="/A334056/b334056.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. 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 G.f.: Sum_{j>=0} (3*j)! * y^j / (j! * 6^j * (1+(1-z)*y)^(3*j+1)).

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

%e       1;

%e       0,    1;

%e       7,    2,   1;

%e     219,   53,   7,  1;

%e   12861, 2296, 226, 16, 1;

%e   ...

%e 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 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 (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 Row sums are A025035.

%Y Cf. A079267, A334057, A334058, A325753.

%K nonn,tabl

%O 0,4

%A _Donovan Young_, Apr 15 2020