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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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.
Sequence in context: A040960 A040946 A040945 * A040947 A040948 A070656
KEYWORD
nonn,tabl
AUTHOR
Donovan Young, Apr 15 2020
STATUS
approved

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Last modified March 2 07:05 EST 2024. Contains 370460 sequences. (Running on oeis4.)