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A334640
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a(n) is the total number of down steps between the 2nd and 3rd up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.
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4
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0, 0, 9, 19, 72, 324, 1595, 8307, 44982, 250648, 1427679, 8274825, 48644310, 289334160, 1738043892, 10529070020, 64252519830, 394601627376, 2437058926871, 15126463230165, 94306717535940, 590318477063700, 3708527622652755, 23374587898663155, 147770791807427880
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OFFSET
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0,3
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COMMENTS
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For n = 2, there is no 3rd up step, a(2) = 9 enumerates the total number of down steps between the 2nd up step and the end of the path.
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LINKS
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FORMULA
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a(0) = a(1) = 0 and a(n) = 2*Sum_{j=1..2} binomial(3*j+1,j) * binomial(3*(n-j),n-j) / ((3*j+1)*(n-j+1)) for n > 1.
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EXAMPLE
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For n = 2, there are the 2-Dyck paths UUDDDD, UDUDDD, UDDUDD. Between the 2nd up step and the end of the path there are a(2) = 4 + 3 + 2 = 9 down steps in total.
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MAPLE
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b:= proc(x, y, u, c) option remember; `if`(x=0, c,
`if`(y+2<x, b(x-1, y+2, min(u+1, 3), c), 0)+
`if`(y>0, b(x-1, y-1, u, c+`if`(u=2, 1, 0)), 0))
end:
a:= n-> b(3*n, 0$3):
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0$2, 9][n+1],
(3*(n-1)*(3*n-8)*(3*n-7)*(13*n-20)*a(n-1))/
(2*(13*n-33)*(n-2)*(2*n-3)*n))
end:
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MATHEMATICA
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a[0] = a[1] = 0; a[n_] := 2 * Sum[Binomial[3*j + 1, j] * Binomial[3*(n - j), n - j]/((3*j + 1)*(n - j + 1)), {j, 1, 2}]; Array[a, 25, 0] (* Amiram Eldar, May 09 2020 *)
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PROG
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(PARI) a(n) = if (n<=1, 0, 2*sum(j=1, 2, binomial(3*j+1, j) * binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)))); \\ Michel Marcus, May 09 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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