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A334682
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a(n) is the total number of down-steps after the final up-step in all 3-Dyck paths of length 4*n (n up-steps and 3*n down-steps).
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10
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0, 3, 18, 118, 829, 6115, 46736, 366912, 2941528, 23981628, 198224910, 1657364566, 13992405626, 119118427610, 1021399476720, 8813544248100, 76475285228304, 666865500290884, 5840843616021192, 51361847992315320, 453282040123194425, 4013440075484640675
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OFFSET
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0,2
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COMMENTS
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A 3-Dyck path is a lattice path with steps U = (1, 3), d = (1, -1) that starts at (0,0), stays (weakly) above the x-axis, and ends at the x-axis.
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LINKS
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FORMULA
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a(n) = binomial(4*(n+1)+1, n+1)/(4*(n+1)+1) - binomial(4*n+1, n)/(4*n+1).
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EXAMPLE
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For n=2 the a(2)=18 is the total number of down-steps after the last up-step in UdddUddd, UddUdddd, UdUddddd, UUdddddd.
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MAPLE
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b:= proc(x, y) option remember; `if`(x=y, x,
`if`(y+3<x, b(x-1, y+3), 0)+`if`(y>0, b(x-1, y-1), 0))
end:
a:= n-> b(4*n, 0):
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 3*n, (8*(4*n-1)*
(2*n-1)*(4*n-3)*n*(229*n^2+303*n+98)*a(n-1))/
(3*(n-1)*(3*n+2)*(3*n+4)*(n+1)*(229*n^2-155*n+24)))
end:
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MATHEMATICA
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nmax = 21;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^4 + O[x]^(nmax + 2), nmax + 2];
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PROG
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(PARI) a(n) = {binomial(4*(n+1)+1, n+1)/(4*(n+1)+1) - binomial(4*n+1, n)/(4*n+1)} \\ Andrew Howroyd, May 08 2020
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CROSSREFS
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First order differences of A002293.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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