A334057 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 4n.

1, 0, 1, 31, 3, 1, 5474, 288, 12, 1, 2554091, 72026, 1476, 31, 1, 2502018819, 43635625, 508610, 5505, 65, 1, 4456194509950, 52673302074, 394246455, 2559565, 16710, 120, 1, 13077453070386914, 111562882654972, 580589062179, 2504572910, 10288390, 43806, 203, 1

Offset

0,4

Comments

In this generalized game of memory n indistinguishable quadruples of matched cards are placed on the vertices of the path of length 4n. A polyomino is a quadruple on four adjacent vertices.

T(n,k) is the number of set partitions of {1..4n} into n sets of 4 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} (4*j)! * y^j / (j! * 24^j * (1+(1-z)*y)^(4*j+1)).

T(n,k) = Sum_{j=0..n-k} (-1)^(n-j-k)*(n+3*j)!/(24^j*j!*(n-j-k)!*k!). - Andrew Howroyd, Apr 16 2020

Example

The first few rows of T(n,k) are:
        1;
        0,     1;
       31,     3,    1;
     5474,   288,   12,  1;
  2554091, 72026, 1476, 31, 1;
  ...
For n=2 and k=1 the polyomino must start either on the second vertex of the path, the third, or the fourth, otherwise the remaining quadruple will also form a polyomino; thus T(2,1) = 3.

Mathematica

CoefficientList[Normal[Series[Sum[y^j*(4*j)!/24^j/j!/(1+y*(1-z))^(4*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+3*j)!/(24^j*j!*(n-j-k)!*k!))} \\ Andrew Howroyd, Apr 16 2020

Crossrefs

Row sums are A025036.

Cf. A079267, A334056, A334058, A325753.

Sequence in context: A040960 A040946 A040945 * A040947 A040948 A070656

Adjacent sequences: A334054 A334055 A334056 * A334058 A334059 A334060

Keyword

nonn,tabl

Author

Donovan Young, Apr 15 2020

Status

approved