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 A334061 Triangle read by rows: T(n,k) is the number of set partitions of {1..4n} into n sets of 4 with k disjoint strings of adjacent sets, each being a contiguous set of elements 0
 1, 0, 1, 31, 4, 0, 5474, 292, 9, 0, 2554091, 72318, 1206, 10, 0, 2502018819, 43707943, 438987, 2871, 5, 0, 4456194509950, 52717010017, 351487598, 1622954, 4355, 1, 0, 13077453070386914, 111615599664989, 528618296314, 1764575884, 4080889, 4385, 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 4n, 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} (4*j)! * y^j * (1-(1-z)*y)^(4*j+1) / (j! * 24^j * (1-(1-z)*y^2)^(4*j+1)). EXAMPLE Triangle begins:         1;         0,     1;        31,     4,   0;      5474,   292,   9,  0;   2554091, 72318,1206, 10, 0;   ... For n=2 and k=1 the configurations are (1,6,7,8),(2,3,4,5), as well as (1,2,7,8),(3,4,5,6) and also (1,2,3,8),(4,5,6,7) (i.e. configurations with a single contiguous set) and (1,2,3,4),(5,6,7,8) (i.e. two adjacent contiguous sets); hence T(2,1) = 4. MATHEMATICA CoefficientList[Normal[Series[Sum[y^j*(4*j)!/24^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(4*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}] PROG (PARI) T(n)={my(v=Vec(sum(j=0, n, (4*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(4*j+1) / (j! * 24^j * (1-(1-y)*x^2 + O(x*x^n))^(4*j+1))))); vector(#v, i, Vecrev(v[i], i))} { my(A=T(8)); for(n=1, #A, print(A[n])) } CROSSREFS Row sums are A025036. Column k=0 is column 0 of A334057. Cf. A079267, A334056, A334057, A334058, A334059, A334060, A325753. Sequence in context: A107811 A109837 A128372 * A040942 A153072 A040943 Adjacent sequences:  A334058 A334059 A334060 * A334062 A334063 A334064 KEYWORD nonn,tabl AUTHOR Donovan Young, May 26 2020 STATUS approved

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Last modified September 24 07:28 EDT 2020. Contains 337317 sequences. (Running on oeis4.)