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 A334060 Triangle read by rows: T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k disjoint strings of adjacent sets, each being a contiguous set of elements 1
 1, 0, 1, 7, 3, 0, 219, 56, 5, 0, 12861, 2352, 183, 4, 0, 1215794, 174137, 11145, 323, 1, 0, 169509845, 19970411, 1078977, 30833, 334, 0, 0, 32774737463, 3280250014, 153076174, 4056764, 55379, 206, 0, 0, 8400108766161, 730845033406, 29989041076, 727278456, 10341101, 67730, 70, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Number of configurations with k connected components (consisting of polyomino matchings) in the generalized game of memory played on the path of length 3n, see [Young]. LINKS Donovan Young, Polyomino matchings in generalised games of memory and linear k-chord diagrams, arXiv:2004.06921 [math.CO], 2020. FORMULA G.f.: Sum_{j>=0} (3*j)! * y^j * (1-(1-z)*y)^(3*j+1) / (j! * 6^j * (1-(1-z)*y^2)^(3*j+1)). EXAMPLE Triangle begins:       1;       0,    1;       7,    3,   0;     219,   56,   5, 0;   12861, 2352, 183, 4, 0;   ... For n=2 and k=1 the configurations are (1,5,6),(2,3,4) and (1,2,6),(3,4,5) (i.e. configurations with a single contiguous set) and (1,2,3),(4,5,6) (i.e. two adjacent contiguous sets); hence T(2,1) = 3. MATHEMATICA CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(3*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}] PROG (PARI) T(n)={my(v=Vec(sum(j=0, n, (3*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(3*j+1) / (j! * 6^j * (1-(1-y)*x^2 + O(x*x^n))^(3*j+1))))); vector(#v, i, Vecrev(v[i], i))} { my(A=T(8)); for(n=1, #A, print(A[n])) } CROSSREFS Row sums are A025035. Column k=0 is column 0 of A334056. Cf. A079267, A334056, A334057, A334058, A334059, A325753. Sequence in context: A327574 A253905 A154159 * A083803 A136595 A111475 Adjacent sequences:  A334057 A334058 A334059 * A334061 A334062 A334063 KEYWORD nonn,tabl AUTHOR Donovan Young, May 26 2020 STATUS approved

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Last modified September 20 04:04 EDT 2020. Contains 337264 sequences. (Running on oeis4.)