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A334056
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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 3n.
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6
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1, 0, 1, 7, 2, 1, 219, 53, 7, 1, 12861, 2296, 226, 16, 1, 1215794, 171785, 13080, 710, 30, 1, 169509845, 19796274, 1228655, 53740, 1835, 50, 1, 32774737463, 3260279603, 170725639, 6250755, 178325, 4137, 77, 1, 8400108766161, 727564783392, 32944247308, 1036855344, 25359670, 507584, 8428, 112, 1
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OFFSET
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0,4
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COMMENTS
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In this generalized game of memory n indistinguishable triples of matched cards are placed on the vertices of the path of length 3n. A polyomino is a triple on three adjacent vertices. For dominoes in ordinary memory on the path of length 2n, see A079267.
T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k of the sets being a contiguous set of elements. - Andrew Howroyd, Apr 16 2020
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LINKS
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FORMULA
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G.f.: Sum_{j>=0} (3*j)! * y^j / (j! * 6^j * (1+(1-z)*y)^(3*j+1)).
T(n,k) = Sum_{j=0..n-k} (-1)^(n-j-k)*(n+2*j)!/(6^j*j!*(n-j-k)!*k!). - Andrew Howroyd, Apr 16 2020
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EXAMPLE
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The first few rows of T(n,k) are:
1;
0, 1;
7, 2, 1;
219, 53, 7, 1;
12861, 2296, 226, 16, 1;
...
For n=2 and k=1 the polyomino must start either on the second vertex of the path, or the third, otherwise the remaining triple will also form a polyomino; thus T(2,1) = 2.
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MATHEMATICA
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CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!/(1+y*(1-z))^(3*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]
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PROG
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(PARI) T(n, k)={sum(j=0, n-k, (-1)^(n-j-k)*(n+2*j)!/(6^j*j!*(n-j-k)!*k!))} \\ Andrew Howroyd, Apr 16 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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