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A334609
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a(n) is the total number of down-steps after the final up-step in all 3_2-Dyck paths of length 4*n (n up-steps and 3*n down-steps).
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2
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0, 6, 46, 339, 2553, 19723, 155805, 1253931, 10249096, 84864051, 710429304, 6003238901, 51140131770, 438729741450, 3787208722815, 32871470376123, 286706337100656, 2511620756461504, 22089299382478728, 194966351598215340, 1726424465382128205
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OFFSET
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0,2
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COMMENTS
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A 3_2-Dyck path is a lattice path with steps U = (1, 3), d = (1, -1) that starts at (0,0), stays (weakly) above y = -2, and ends at the x-axis.
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LINKS
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FORMULA
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a(n) = 3*binomial(4*(n+1) + 3, n+1)/(4*(n+1) + 3) - 9*binomial(4*n+3, n)/(4*n + 3).
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EXAMPLE
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For n = 1, a(1) = 6 is the total number of down-steps after the last up-step in Uddd, dUdd, ddUd.
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MATHEMATICA
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a[n_] := 3 * Binomial[4*n + 7, n + 1]/(4*n + 7) - 9 * Binomial[4*n + 3, n]/(4*n + 3); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)
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PROG
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(SageMath) [3*binomial(4*(n + 1) + 3, n + 1)/(4*(n + 1) + 3) - 9*binomial(4*n + 3, n)/(4*n + 3) for n in srange(30)] # Benjamin Hackl, May 13 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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