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
KEYWORD
nonn,tabl
AUTHOR
Donovan Young, Apr 15 2020
STATUS
approved