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A212389 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 9). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

Lengths of descents are unrestricted.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..800

Vaclav Kotesovec, Asymptotic of subsequences of A212382

FORMULA

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^9).

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 = 9 and r = 0.4164039515514120671..., s = 1.882616423435763466... 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

EXAMPLE

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.

MAPLE

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);

CROSSREFS

Column k=9 of A212382.

Sequence in context: A039633 A265451 A180195 * A241074 A020062 A185035

Adjacent sequences:  A212386 A212387 A212388 * A212390 A212391 A212392

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 12 2012

STATUS

approved

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Last modified April 24 20:29 EDT 2019. Contains 322446 sequences. (Running on oeis4.)