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.

1, 0, 1, 7, 2, 1, 219, 53, 7, 1, 12861, 2296, 226, 16, 1, 1215794, 171785, 13080, 710, 30, 1, 169509845, 19796274, 1228655, 53740, 1835, 50, 1, 32774737463, 3260279603, 170725639, 6250755, 178325, 4137, 77, 1, 8400108766161, 727564783392, 32944247308, 1036855344, 25359670, 507584, 8428, 112, 1

Offset: 0

Author

Donovan Young, Apr 15 2020

Keywords

• nonn
• tabl

Comments

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.

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

Examples

			The first few rows of T(n,k) are:
      1;
      0,    1;
      7,    2,   1;
    219,   53,   7,  1;
  12861, 2296, 226, 16, 1;
  ...
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.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020. See also J. Int. Seq., Vol. 23 (2020), Article 20.9.1.

Crossrefs

Row sums are A025035.

Cf. A079267, A334057, A334058, A325753.

Programs

Mathematica

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}]

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

Formula

G.f.: Sum_{j>=0} (3*j)! * y^j / (j! * 6^j * (1+(1-z)*y)^(3*j+1)).

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