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A287987
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Number of Dyck paths of semilength n such that all positive levels have the same number of peaks.
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6
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1, 1, 1, 3, 1, 8, 13, 13, 54, 132, 280, 547, 1219, 3904, 11107, 25082, 53777, 137751, 419831, 1257599, 3453557, 8911341, 22636845, 59890162, 172264224, 529706648, 1630328686, 4765347773, 13125989799, 35253234315, 97531470556, 287880507391, 894915519516
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OFFSET
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0,4
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LINKS
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EXAMPLE
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. a(3) = 3: /\ /\
. /\/\/\ /\/ \ / \/\ .
.
. a(5) = 8:
. /\/\ /\/\ /\/\
. /\/\/\/\/\ /\/\/ \ /\/ \/\ / \/\/\
.
. /\ /\ /\ /\
. /\/ \ / \/\ /\/ \ / \/\
. /\/ \ /\/ \ / \/\ / \/\ .
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MAPLE
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b:= proc(n, k, j) option remember; `if`(n=j, 1,
add(binomial(i, k)*binomial(j-1, i-1-k)
*b(n-j, k, i), i=1+k..min(j+k, n-j)))
end:
a:= n-> 1+add(b(n, j$2), j=1..n/2):
seq(a(n), n=0..33);
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MATHEMATICA
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b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Binomial[i, k]*Binomial[j - 1, i - 1 - k]*b[n - j, k, i], {i, 1 + k, Min[j + k, n - j]}]];
a[n_] := 1 + Sum[b[n, j, j], {j, 1, n/2}];
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CROSSREFS
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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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