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A210735
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Number of Dyck n-paths all of whose ascents and descents have prime lengths.
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2
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1, 0, 1, 1, 1, 4, 2, 10, 10, 22, 46, 64, 167, 245, 560, 1035, 1978, 4210, 7715, 16497, 31929, 65216, 133295, 266244, 553750, 1116404, 2308931, 4738660, 9742795, 20204902, 41622910, 86539105, 179358694, 373018581, 777157221, 1618773690, 3382590684, 7065505631
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OFFSET
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0,6
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^(3/2), where d = 2.1792514215908330337..., c = 0.4751731999905254789... . - Vaclav Kotesovec, Sep 02 2014
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EXAMPLE
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a(0) = 1: the empty path.
a(1) = 0.
a(2) = 1: UUDD.
a(3) = 1: UUUDDD.
a(4) = 1: UUDDUUDD.
a(5) = 4: UUDDUUUDDD, UUUDDDUUDD, UUUDDUUDDD, UUUUUDDDDD.
a(6) = 2: UUDDUUDDUUDD, UUUDDDUUUDDD.
a(7) = 10: UUDDUUDDUUUDDD, UUDDUUUDDDUUDD, UUDDUUUDDUUDDD, UUDDUUUUUDDDDD, UUUDDDUUDDUUDD, UUUDDUUDDDUUDD, UUUDDUUDDUUDDD, UUUUUDDDDDUUDD, UUUUUDDUUDDDDD, UUUUUUUDDDDDDD.
a(8) = 10: UUDDUUDDUUDDUUDD, UUDDUUUDDDUUUDDD, UUUDDDUUDDUUUDDD, UUUDDDUUUDDDUUDD, UUUDDDUUUDDUUDDD, UUUDDDUUUUUDDDDD, UUUDDUUDDDUUUDDD, UUUDDUUUDDDUUDDD, UUUUUDDDDDUUUDDD, UUUUUDDDUUUDDDDD.
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MAPLE
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with(numtheory):
b:= proc(x, y, u) option remember;
`if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, `if`(u,
add(b(x-ithprime(t), y, false), t=1..pi(x)),
add(b(x, y-ithprime(t), true ), t=1..pi(y)))))
end:
a:= n-> b(n$2, true):
seq(a(n), n=0..40);
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MATHEMATICA
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b[x_, y_, u_] := b[x, y, u] = If[x<0 || y<x, 0, If[x==0 && y==0, 1, If[u, Sum[b[x-Prime[t], y, False], {t, 1, PrimePi[x]}], Sum[b[x, y-Prime[t], True], {t, 1, PrimePi[y]}]]]];
a[n_] := b[n, n, True];
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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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