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.

1, 0, 1, 121, 4, 1, 124760, 1347, 18, 1, 486854621, 2001548, 8154, 52, 1, 5184423824705, 10231953233, 17045774, 35542, 121, 1, 123243726413573515, 134835947255262, 112619668659, 102416812, 124881, 246, 1, 5717986519188343198259, 3821094862609800013, 1820735766620673, 863827126967, 486979381, 375627, 455, 1

Offset: 0

Author

Donovan Young, Apr 15 2020

Keywords

• nonn
• tabl

Comments

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.

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

Examples

			The first few rows of T(n,k) are:
          1;
          0,       1;
        121,       4,    1;
     124760,    1347,   18,  1;
  486854621, 2001548, 8154, 52, 1;
  ...
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.
		

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.

Crossrefs

Row sums are A025037.

Cf. A079267, A334056, A334057, A325753.

Programs

Mathematica

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}]

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

Formula

G.f.: Sum_{j>=0} (5*j)! * y^j / (j! * 120^j * (1+(1-z)*y)^(5*j+1)).

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