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A212390
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Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 10).
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2
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 13, 79, 365, 1366, 4369, 12377, 31825, 75583, 167961, 352718, 705466, 1352585, 2501205, 4495351, 7956391, 14221936, 26802361, 56058016, 133316626, 350785307, 967683665, 2677259721, 7246005881, 18977267621, 47931495649
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OFFSET
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0,12
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COMMENTS
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Lengths of descents are unrestricted.
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LINKS
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FORMULA
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G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^10).
a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 10 and r = 0.421937635689419083..., s = 1.885352542104400040... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014
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EXAMPLE
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a(0) = 1: the empty path.
a(1) = 1: UD.
a(11) = 2: UDUDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUUDDDDDDDDDDD.
a(12) = 13: UDUDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUUDDDDDDDDDDD, UUUUUUUUUUUDDDDDDDDDDDUD, UUUUUUUUUUUDDDDDDDDDDUDD, UUUUUUUUUUUDDDDDDDDDUDDD, UUUUUUUUUUUDDDDDDDDUDDDD, UUUUUUUUUUUDDDDDDDUDDDDD, UUUUUUUUUUUDDDDDDUDDDDDD, UUUUUUUUUUUDDDDDUDDDDDDD, UUUUUUUUUUUDDDDUDDDDDDDD, UUUUUUUUUUUDDDUDDDDDDDDD, UUUUUUUUUUUDDUDDDDDDDDDD, UUUUUUUUUUUDUDDDDDDDDDDD.
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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-(10*t+1), y, false), t=0..(x-1)/10), 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)^10), A), x, n+1), x, n):
seq(a(n), n=0..40);
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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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