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A212388 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 8). 2
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 11, 56, 221, 716, 2003, 5006, 11441, 24312, 48648, 92721, 170811, 311886, 589590, 1220979, 2864973, 7450852, 20309628, 55305706, 146505451, 373452808, 913836082, 2150455648, 4887179761, 10794337952, 23375638064, 50219351232 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Lengths of descents are unrestricted.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..750

Vaclav Kotesovec, Asymptotic of subsequences of A212382

FORMULA

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^8).

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 = 8 and r = 0.4098875088359862102..., s = 1.880071788712472133... 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(9) = 2: UDUDUDUDUDUDUDUDUD, UUUUUUUUUDDDDDDDDD.

a(10) = 11: UDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUDDDDDDDDD, UUUUUUUUUDDDDDDDDDUD, UUUUUUUUUDDDDDDDDUDD, UUUUUUUUUDDDDDDDUDDD, UUUUUUUUUDDDDDDUDDDD, UUUUUUUUUDDDDDUDDDDD, UUUUUUUUUDDDDUDDDDDD, UUUUUUUUUDDDUDDDDDDD, UUUUUUUUUDDUDDDDDDDD, UUUUUUUUUDUDDDDDDDDD.

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-(8*t+1), y, false), t=0..(x-1)/8), 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)^8), A), x, n+1), x, n):

seq(a(n), n=0..40);

CROSSREFS

Column k=8 of A212382.

Sequence in context: A115205 A306753 A306860 * A198769 A037554 A106804

Adjacent sequences:  A212385 A212386 A212387 * A212389 A212390 A212391

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 12 2012

STATUS

approved

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Last modified April 19 06:59 EDT 2019. Contains 322237 sequences. (Running on oeis4.)