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A212389
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Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 9).
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2
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 12, 67, 287, 1002, 3004, 8009, 19449, 43759, 92380, 184787, 353137, 650497, 1170632, 2110021, 3977161, 8271836, 19536661, 51111062, 140210129, 385123916, 1032218316, 2670065961, 6645249777, 15922990909, 36823807747, 82485177457
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OFFSET
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0,11
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COMMENTS
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Lengths of descents are unrestricted.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..800
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FORMULA
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G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^9).
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EXAMPLE
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a(0) = 1: the empty path.
a(1) = 1: UD.
a(10) = 2: UDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUDDDDDDDDDD.
a(11) = 12: UDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUDDDDDDDDDD, UUUUUUUUUUDDDDDDDDDDUD, UUUUUUUUUUDDDDDDDDDUDD, UUUUUUUUUUDDDDDDDDUDDD, UUUUUUUUUUDDDDDDDUDDDD, UUUUUUUUUUDDDDDDUDDDDD, UUUUUUUUUUDDDDDUDDDDDD, UUUUUUUUUUDDDDUDDDDDDD, UUUUUUUUUUDDDUDDDDDDDD, UUUUUUUUUUDDUDDDDDDDDD, UUUUUUUUUUDUDDDDDDDDDD.
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MAPLE
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b:= proc(x, y, u) option remember;
`if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
`if`(u, add (b(x-(9*t+1), y, false), t=0..(x-1)/9), 0)))
end:
a:= n-> b(n$2, true):
seq (a(n), n=0..40);
# second Maple program
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^9), A), x, n+1), x, n):
seq (a(n), n=0..40);
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CROSSREFS
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Column k=9 of A212382.
Sequence in context: A197745 A039633 A180195 * A020062 A185035 A185235
Adjacent sequences: A212386 A212387 A212388 * A212390 A212391 A212392
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, May 12 2012
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STATUS
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approved
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