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

0,4

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

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.

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

Example

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.

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}]

Prog

(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

Crossrefs

Row sums are A025037.

Cf. A079267, A334056, A334057, A325753.

Sequence in context: A079842 A014756 A014748 * A197817 A152144 A282344

Adjacent sequences: A334055 A334056 A334057 * A334059 A334060 A334061

Keyword

nonn,tabl

Author

Donovan Young, Apr 15 2020

Status

approved